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Theorem resf1o 29371
Description: Restriction of functions to a superset of their support creates a bijection. (Contributed by Thierry Arnoux, 12-Sep-2017.)
Hypotheses
Ref Expression
resf1o.1 𝑋 = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶}
resf1o.2 𝐹 = (𝑓𝑋 ↦ (𝑓𝐶))
Assertion
Ref Expression
resf1o (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → 𝐹:𝑋1-1-onto→(𝐵𝑚 𝐶))
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝐶,𝑓   𝑓,𝑉   𝑓,𝑊   𝑓,𝑋   𝑓,𝑍
Allowed substitution hint:   𝐹(𝑓)

Proof of Theorem resf1o
Dummy variables 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resf1o.2 . 2 𝐹 = (𝑓𝑋 ↦ (𝑓𝐶))
2 resexg 5406 . . 3 (𝑓𝑋 → (𝑓𝐶) ∈ V)
32adantl 482 . 2 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ 𝑓𝑋) → (𝑓𝐶) ∈ V)
4 simpr 477 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑔 ∈ (𝐵𝑚 𝐶)) → 𝑔 ∈ (𝐵𝑚 𝐶))
5 difexg 4773 . . . . . . 7 (𝐴𝑉 → (𝐴𝐶) ∈ V)
653ad2ant1 1080 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝐴) → (𝐴𝐶) ∈ V)
7 snex 4874 . . . . . 6 {𝑍} ∈ V
8 xpexg 6920 . . . . . 6 (((𝐴𝐶) ∈ V ∧ {𝑍} ∈ V) → ((𝐴𝐶) × {𝑍}) ∈ V)
96, 7, 8sylancl 693 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝐴) → ((𝐴𝐶) × {𝑍}) ∈ V)
109adantr 481 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑔 ∈ (𝐵𝑚 𝐶)) → ((𝐴𝐶) × {𝑍}) ∈ V)
11 unexg 6919 . . . 4 ((𝑔 ∈ (𝐵𝑚 𝐶) ∧ ((𝐴𝐶) × {𝑍}) ∈ V) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) ∈ V)
124, 10, 11syl2anc 692 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑔 ∈ (𝐵𝑚 𝐶)) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) ∈ V)
1312adantlr 750 . 2 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ 𝑔 ∈ (𝐵𝑚 𝐶)) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) ∈ V)
14 resf1o.1 . . . . 5 𝑋 = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶}
1514rabeq2i 3186 . . . 4 (𝑓𝑋 ↔ (𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶))
1615anbi1i 730 . . 3 ((𝑓𝑋𝑔 = (𝑓𝐶)) ↔ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶)))
17 simprr 795 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑔 = (𝑓𝐶))
18 simprll 801 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 ∈ (𝐵𝑚 𝐴))
19 elmapi 7831 . . . . . . . . 9 (𝑓 ∈ (𝐵𝑚 𝐴) → 𝑓:𝐴𝐵)
2018, 19syl 17 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓:𝐴𝐵)
21 simp3 1061 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊𝐶𝐴) → 𝐶𝐴)
2221ad2antrr 761 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝐶𝐴)
2320, 22fssresd 6033 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓𝐶):𝐶𝐵)
24 simp2 1060 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊𝐶𝐴) → 𝐵𝑊)
25 simp1 1059 . . . . . . . . . 10 ((𝐴𝑉𝐵𝑊𝐶𝐴) → 𝐴𝑉)
2625, 21ssexd 4770 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊𝐶𝐴) → 𝐶 ∈ V)
27 elmapg 7822 . . . . . . . . 9 ((𝐵𝑊𝐶 ∈ V) → ((𝑓𝐶) ∈ (𝐵𝑚 𝐶) ↔ (𝑓𝐶):𝐶𝐵))
2824, 26, 27syl2anc 692 . . . . . . . 8 ((𝐴𝑉𝐵𝑊𝐶𝐴) → ((𝑓𝐶) ∈ (𝐵𝑚 𝐶) ↔ (𝑓𝐶):𝐶𝐵))
2928ad2antrr 761 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → ((𝑓𝐶) ∈ (𝐵𝑚 𝐶) ↔ (𝑓𝐶):𝐶𝐵))
3023, 29mpbird 247 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓𝐶) ∈ (𝐵𝑚 𝐶))
3117, 30eqeltrd 2698 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑔 ∈ (𝐵𝑚 𝐶))
32 undif 4026 . . . . . . . . . . 11 (𝐶𝐴 ↔ (𝐶 ∪ (𝐴𝐶)) = 𝐴)
3332biimpi 206 . . . . . . . . . 10 (𝐶𝐴 → (𝐶 ∪ (𝐴𝐶)) = 𝐴)
3433reseq2d 5361 . . . . . . . . 9 (𝐶𝐴 → (𝑓 ↾ (𝐶 ∪ (𝐴𝐶))) = (𝑓𝐴))
3522, 34syl 17 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 ↾ (𝐶 ∪ (𝐴𝐶))) = (𝑓𝐴))
36 ffn 6007 . . . . . . . . 9 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
37 fnresdm 5963 . . . . . . . . 9 (𝑓 Fn 𝐴 → (𝑓𝐴) = 𝑓)
3820, 36, 373syl 18 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓𝐴) = 𝑓)
3935, 38eqtr2d 2656 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 = (𝑓 ↾ (𝐶 ∪ (𝐴𝐶))))
40 resundi 5374 . . . . . . 7 (𝑓 ↾ (𝐶 ∪ (𝐴𝐶))) = ((𝑓𝐶) ∪ (𝑓 ↾ (𝐴𝐶)))
4139, 40syl6eq 2671 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 = ((𝑓𝐶) ∪ (𝑓 ↾ (𝐴𝐶))))
4217eqcomd 2627 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓𝐶) = 𝑔)
43 simprlr 802 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶)
4425ad2antrr 761 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝐴𝑉)
45 simplr 791 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑍𝐵)
46 eqid 2621 . . . . . . . . . . 11 (𝐵 ∖ {𝑍}) = (𝐵 ∖ {𝑍})
4746ffs2 29369 . . . . . . . . . 10 ((𝐴𝑉𝑍𝐵𝑓:𝐴𝐵) → (𝑓 supp 𝑍) = (𝑓 “ (𝐵 ∖ {𝑍})))
4844, 45, 20, 47syl3anc 1323 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 supp 𝑍) = (𝑓 “ (𝐵 ∖ {𝑍})))
49 sseqin2 3800 . . . . . . . . . . 11 (𝐶𝐴 ↔ (𝐴𝐶) = 𝐶)
5049biimpi 206 . . . . . . . . . 10 (𝐶𝐴 → (𝐴𝐶) = 𝐶)
5122, 50syl 17 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝐴𝐶) = 𝐶)
5243, 48, 513sstr4d 3632 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 supp 𝑍) ⊆ (𝐴𝐶))
53 simpl 473 . . . . . . . . . . . 12 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → 𝑓 ∈ (𝐵𝑚 𝐴))
5453, 19, 363syl 18 . . . . . . . . . . 11 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → 𝑓 Fn 𝐴)
55 inundif 4023 . . . . . . . . . . . 12 ((𝐴𝐶) ∪ (𝐴𝐶)) = 𝐴
5655fneq2i 5949 . . . . . . . . . . 11 (𝑓 Fn ((𝐴𝐶) ∪ (𝐴𝐶)) ↔ 𝑓 Fn 𝐴)
5754, 56sylibr 224 . . . . . . . . . 10 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → 𝑓 Fn ((𝐴𝐶) ∪ (𝐴𝐶)))
58 vex 3192 . . . . . . . . . . 11 𝑓 ∈ V
5958a1i 11 . . . . . . . . . 10 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → 𝑓 ∈ V)
60 simpr 477 . . . . . . . . . 10 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → 𝑍𝐵)
61 inindif 29224 . . . . . . . . . . 11 ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅
6261a1i 11 . . . . . . . . . 10 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅)
63 fnsuppres 7274 . . . . . . . . . 10 ((𝑓 Fn ((𝐴𝐶) ∪ (𝐴𝐶)) ∧ (𝑓 ∈ V ∧ 𝑍𝐵) ∧ ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅) → ((𝑓 supp 𝑍) ⊆ (𝐴𝐶) ↔ (𝑓 ↾ (𝐴𝐶)) = ((𝐴𝐶) × {𝑍})))
6457, 59, 60, 62, 63syl121anc 1328 . . . . . . . . 9 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → ((𝑓 supp 𝑍) ⊆ (𝐴𝐶) ↔ (𝑓 ↾ (𝐴𝐶)) = ((𝐴𝐶) × {𝑍})))
6518, 45, 64syl2anc 692 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → ((𝑓 supp 𝑍) ⊆ (𝐴𝐶) ↔ (𝑓 ↾ (𝐴𝐶)) = ((𝐴𝐶) × {𝑍})))
6652, 65mpbid 222 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 ↾ (𝐴𝐶)) = ((𝐴𝐶) × {𝑍}))
6742, 66uneq12d 3751 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → ((𝑓𝐶) ∪ (𝑓 ↾ (𝐴𝐶))) = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
6841, 67eqtrd 2655 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
6931, 68jca 554 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍}))))
7024ad2antrr 761 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝐵𝑊)
7125ad2antrr 761 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝐴𝑉)
72 elmapi 7831 . . . . . . . . 9 (𝑔 ∈ (𝐵𝑚 𝐶) → 𝑔:𝐶𝐵)
7372ad2antrl 763 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑔:𝐶𝐵)
74 simplr 791 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑍𝐵)
75 fconst6g 6056 . . . . . . . . 9 (𝑍𝐵 → ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶𝐵)
7674, 75syl 17 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶𝐵)
77 disjdif 4017 . . . . . . . . 9 (𝐶 ∩ (𝐴𝐶)) = ∅
7877a1i 11 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝐶 ∩ (𝐴𝐶)) = ∅)
79 fun2 6029 . . . . . . . 8 (((𝑔:𝐶𝐵 ∧ ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶𝐵) ∧ (𝐶 ∩ (𝐴𝐶)) = ∅) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})):(𝐶 ∪ (𝐴𝐶))⟶𝐵)
8073, 76, 78, 79syl21anc 1322 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})):(𝐶 ∪ (𝐴𝐶))⟶𝐵)
81 simprr 795 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
8281eqcomd 2627 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) = 𝑓)
8321ad2antrr 761 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝐶𝐴)
8483, 33syl 17 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝐶 ∪ (𝐴𝐶)) = 𝐴)
8582, 84feq12d 5995 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍})):(𝐶 ∪ (𝐴𝐶))⟶𝐵𝑓:𝐴𝐵))
8680, 85mpbid 222 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑓:𝐴𝐵)
87 elmapg 7822 . . . . . . 7 ((𝐵𝑊𝐴𝑉) → (𝑓 ∈ (𝐵𝑚 𝐴) ↔ 𝑓:𝐴𝐵))
8887biimpar 502 . . . . . 6 (((𝐵𝑊𝐴𝑉) ∧ 𝑓:𝐴𝐵) → 𝑓 ∈ (𝐵𝑚 𝐴))
8970, 71, 86, 88syl21anc 1322 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑓 ∈ (𝐵𝑚 𝐴))
9071, 74, 86, 47syl3anc 1323 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓 supp 𝑍) = (𝑓 “ (𝐵 ∖ {𝑍})))
9181adantr 481 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
9291fveq1d 6155 . . . . . . . 8 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (𝑓𝑥) = ((𝑔 ∪ ((𝐴𝐶) × {𝑍}))‘𝑥))
9373adantr 481 . . . . . . . . . 10 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑔:𝐶𝐵)
94 ffn 6007 . . . . . . . . . 10 (𝑔:𝐶𝐵𝑔 Fn 𝐶)
9593, 94syl 17 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑔 Fn 𝐶)
96 fconstg 6054 . . . . . . . . . . 11 (𝑍𝐵 → ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶{𝑍})
9796ad3antlr 766 . . . . . . . . . 10 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶{𝑍})
98 ffn 6007 . . . . . . . . . 10 (((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶{𝑍} → ((𝐴𝐶) × {𝑍}) Fn (𝐴𝐶))
9997, 98syl 17 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → ((𝐴𝐶) × {𝑍}) Fn (𝐴𝐶))
10077a1i 11 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (𝐶 ∩ (𝐴𝐶)) = ∅)
101 simpr 477 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑥 ∈ (𝐴𝐶))
102 fvun2 6232 . . . . . . . . 9 ((𝑔 Fn 𝐶 ∧ ((𝐴𝐶) × {𝑍}) Fn (𝐴𝐶) ∧ ((𝐶 ∩ (𝐴𝐶)) = ∅ ∧ 𝑥 ∈ (𝐴𝐶))) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍}))‘𝑥) = (((𝐴𝐶) × {𝑍})‘𝑥))
10395, 99, 100, 101, 102syl112anc 1327 . . . . . . . 8 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍}))‘𝑥) = (((𝐴𝐶) × {𝑍})‘𝑥))
104 fvconst 6391 . . . . . . . . 9 ((((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶{𝑍} ∧ 𝑥 ∈ (𝐴𝐶)) → (((𝐴𝐶) × {𝑍})‘𝑥) = 𝑍)
10597, 101, 104syl2anc 692 . . . . . . . 8 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (((𝐴𝐶) × {𝑍})‘𝑥) = 𝑍)
10692, 103, 1053eqtrd 2659 . . . . . . 7 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (𝑓𝑥) = 𝑍)
10786, 106suppss 7277 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓 supp 𝑍) ⊆ 𝐶)
10890, 107eqsstr3d 3624 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶)
10981reseq1d 5360 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓𝐶) = ((𝑔 ∪ ((𝐴𝐶) × {𝑍})) ↾ 𝐶))
110 res0 5365 . . . . . . . . . 10 (((𝐴𝐶) × {𝑍}) ↾ ∅) = ∅
111 res0 5365 . . . . . . . . . 10 (𝑔 ↾ ∅) = ∅
112110, 111eqtr4i 2646 . . . . . . . . 9 (((𝐴𝐶) × {𝑍}) ↾ ∅) = (𝑔 ↾ ∅)
11377reseq2i 5358 . . . . . . . . 9 (((𝐴𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴𝐶))) = (((𝐴𝐶) × {𝑍}) ↾ ∅)
11477reseq2i 5358 . . . . . . . . 9 (𝑔 ↾ (𝐶 ∩ (𝐴𝐶))) = (𝑔 ↾ ∅)
115112, 113, 1143eqtr4ri 2654 . . . . . . . 8 (𝑔 ↾ (𝐶 ∩ (𝐴𝐶))) = (((𝐴𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴𝐶)))
116115a1i 11 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑔 ↾ (𝐶 ∩ (𝐴𝐶))) = (((𝐴𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴𝐶))))
117 fresaunres1 6039 . . . . . . 7 ((𝑔:𝐶𝐵 ∧ ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶𝐵 ∧ (𝑔 ↾ (𝐶 ∩ (𝐴𝐶))) = (((𝐴𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴𝐶)))) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍})) ↾ 𝐶) = 𝑔)
11873, 76, 116, 117syl3anc 1323 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍})) ↾ 𝐶) = 𝑔)
119109, 118eqtr2d 2656 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑔 = (𝑓𝐶))
12089, 108, 119jca31 556 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶)))
12169, 120impbida 876 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → (((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶)) ↔ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))))
12216, 121syl5bb 272 . 2 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → ((𝑓𝑋𝑔 = (𝑓𝐶)) ↔ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))))
1231, 3, 13, 122f1od 6845 1 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → 𝐹:𝑋1-1-onto→(𝐵𝑚 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3189  cdif 3556  cun 3557  cin 3558  wss 3559  c0 3896  {csn 4153  cmpt 4678   × cxp 5077  ccnv 5078  cres 5081  cima 5082   Fn wfn 5847  wf 5848  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610   supp csupp 7247  𝑚 cmap 7809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-supp 7248  df-map 7811
This theorem is referenced by:  eulerpartgbij  30239
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