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Theorem resf1o 30453
 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 5871 . . 3 (𝑓𝑋 → (𝑓𝐶) ∈ V)
32adantl 485 . 2 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ 𝑓𝑋) → (𝑓𝐶) ∈ V)
4 simpr 488 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑔 ∈ (𝐵m 𝐶)) → 𝑔 ∈ (𝐵m 𝐶))
5 difexg 5204 . . . . . . 7 (𝐴𝑉 → (𝐴𝐶) ∈ V)
653ad2ant1 1130 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝐴) → (𝐴𝐶) ∈ V)
7 snex 5305 . . . . . 6 {𝑍} ∈ V
8 xpexg 7448 . . . . . 6 (((𝐴𝐶) ∈ V ∧ {𝑍} ∈ V) → ((𝐴𝐶) × {𝑍}) ∈ V)
96, 7, 8sylancl 589 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝐴) → ((𝐴𝐶) × {𝑍}) ∈ V)
109adantr 484 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑔 ∈ (𝐵m 𝐶)) → ((𝐴𝐶) × {𝑍}) ∈ V)
11 unexg 7447 . . . 4 ((𝑔 ∈ (𝐵m 𝐶) ∧ ((𝐴𝐶) × {𝑍}) ∈ V) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) ∈ V)
124, 10, 11syl2anc 587 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑔 ∈ (𝐵m 𝐶)) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) ∈ V)
1312adantlr 714 . 2 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ 𝑔 ∈ (𝐵m 𝐶)) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) ∈ V)
14 resf1o.1 . . . . 5 𝑋 = {𝑓 ∈ (𝐵m 𝐴) ∣ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶}
1514rabeq2i 3464 . . . 4 (𝑓𝑋 ↔ (𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶))
1615anbi1i 626 . . 3 ((𝑓𝑋𝑔 = (𝑓𝐶)) ↔ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶)))
17 simprr 772 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑔 = (𝑓𝐶))
18 simprll 778 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 ∈ (𝐵m 𝐴))
19 elmapi 8403 . . . . . . . . 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 1135 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊𝐶𝐴) → 𝐶𝐴)
2221ad2antrr 725 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝐶𝐴)
2320, 22fssresd 6518 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓𝐶):𝐶𝐵)
24 simp2 1134 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊𝐶𝐴) → 𝐵𝑊)
25 simp1 1133 . . . . . . . . . 10 ((𝐴𝑉𝐵𝑊𝐶𝐴) → 𝐴𝑉)
2625, 21ssexd 5201 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊𝐶𝐴) → 𝐶 ∈ V)
27 elmapg 8394 . . . . . . . . 9 ((𝐵𝑊𝐶 ∈ V) → ((𝑓𝐶) ∈ (𝐵m 𝐶) ↔ (𝑓𝐶):𝐶𝐵))
2824, 26, 27syl2anc 587 . . . . . . . 8 ((𝐴𝑉𝐵𝑊𝐶𝐴) → ((𝑓𝐶) ∈ (𝐵m 𝐶) ↔ (𝑓𝐶):𝐶𝐵))
2928ad2antrr 725 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → ((𝑓𝐶) ∈ (𝐵m 𝐶) ↔ (𝑓𝐶):𝐶𝐵))
3023, 29mpbird 260 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓𝐶) ∈ (𝐵m 𝐶))
3117, 30eqeltrd 2912 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑔 ∈ (𝐵m 𝐶))
32 undif 4403 . . . . . . . . . . 11 (𝐶𝐴 ↔ (𝐶 ∪ (𝐴𝐶)) = 𝐴)
3332biimpi 219 . . . . . . . . . 10 (𝐶𝐴 → (𝐶 ∪ (𝐴𝐶)) = 𝐴)
3433reseq2d 5826 . . . . . . . . 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 6487 . . . . . . . . 9 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
37 fnresdm 6439 . . . . . . . . 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 2857 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 = (𝑓 ↾ (𝐶 ∪ (𝐴𝐶))))
40 resundi 5840 . . . . . . 7 (𝑓 ↾ (𝐶 ∪ (𝐴𝐶))) = ((𝑓𝐶) ∪ (𝑓 ↾ (𝐴𝐶)))
4139, 40syl6eq 2872 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 = ((𝑓𝐶) ∪ (𝑓 ↾ (𝐴𝐶))))
4217eqcomd 2827 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓𝐶) = 𝑔)
43 simprlr 779 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶)
4425ad2antrr 725 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝐴𝑉)
45 simplr 768 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑍𝐵)
46 eqid 2821 . . . . . . . . . . 11 (𝐵 ∖ {𝑍}) = (𝐵 ∖ {𝑍})
4746ffs2 30451 . . . . . . . . . 10 ((𝐴𝑉𝑍𝐵𝑓:𝐴𝐵) → (𝑓 supp 𝑍) = (𝑓 “ (𝐵 ∖ {𝑍})))
4844, 45, 20, 47syl3anc 1368 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 supp 𝑍) = (𝑓 “ (𝐵 ∖ {𝑍})))
49 sseqin2 4167 . . . . . . . . . . 11 (𝐶𝐴 ↔ (𝐴𝐶) = 𝐶)
5049biimpi 219 . . . . . . . . . 10 (𝐶𝐴 → (𝐴𝐶) = 𝐶)
5122, 50syl 17 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝐴𝐶) = 𝐶)
5243, 48, 513sstr4d 3990 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 supp 𝑍) ⊆ (𝐴𝐶))
53 simpl 486 . . . . . . . . . . . 12 ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑍𝐵) → 𝑓 ∈ (𝐵m 𝐴))
5453, 19, 363syl 18 . . . . . . . . . . 11 ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑍𝐵) → 𝑓 Fn 𝐴)
55 inundif 4400 . . . . . . . . . . . 12 ((𝐴𝐶) ∪ (𝐴𝐶)) = 𝐴
5655fneq2i 6424 . . . . . . . . . . 11 (𝑓 Fn ((𝐴𝐶) ∪ (𝐴𝐶)) ↔ 𝑓 Fn 𝐴)
5754, 56sylibr 237 . . . . . . . . . 10 ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑍𝐵) → 𝑓 Fn ((𝐴𝐶) ∪ (𝐴𝐶)))
58 vex 3474 . . . . . . . . . . 11 𝑓 ∈ V
5958a1i 11 . . . . . . . . . 10 ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑍𝐵) → 𝑓 ∈ V)
60 simpr 488 . . . . . . . . . 10 ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑍𝐵) → 𝑍𝐵)
61 inindif 30265 . . . . . . . . . . 11 ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅
6261a1i 11 . . . . . . . . . 10 ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑍𝐵) → ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅)
63 fnsuppres 7832 . . . . . . . . . 10 ((𝑓 Fn ((𝐴𝐶) ∪ (𝐴𝐶)) ∧ (𝑓 ∈ V ∧ 𝑍𝐵) ∧ ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅) → ((𝑓 supp 𝑍) ⊆ (𝐴𝐶) ↔ (𝑓 ↾ (𝐴𝐶)) = ((𝐴𝐶) × {𝑍})))
6457, 59, 60, 62, 63syl121anc 1372 . . . . . . . . 9 ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑍𝐵) → ((𝑓 supp 𝑍) ⊆ (𝐴𝐶) ↔ (𝑓 ↾ (𝐴𝐶)) = ((𝐴𝐶) × {𝑍})))
6518, 45, 64syl2anc 587 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → ((𝑓 supp 𝑍) ⊆ (𝐴𝐶) ↔ (𝑓 ↾ (𝐴𝐶)) = ((𝐴𝐶) × {𝑍})))
6652, 65mpbid 235 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 ↾ (𝐴𝐶)) = ((𝐴𝐶) × {𝑍}))
6742, 66uneq12d 4116 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → ((𝑓𝐶) ∪ (𝑓 ↾ (𝐴𝐶))) = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
6841, 67eqtrd 2856 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
6931, 68jca 515 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍}))))
7024ad2antrr 725 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝐵𝑊)
7125ad2antrr 725 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝐴𝑉)
72 elmapi 8403 . . . . . . . . 9 (𝑔 ∈ (𝐵m 𝐶) → 𝑔:𝐶𝐵)
7372ad2antrl 727 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑔:𝐶𝐵)
74 simplr 768 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑍𝐵)
75 fconst6g 6541 . . . . . . . . 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 4394 . . . . . . . . 9 (𝐶 ∩ (𝐴𝐶)) = ∅
7877a1i 11 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝐶 ∩ (𝐴𝐶)) = ∅)
79 fun2 6514 . . . . . . . 8 (((𝑔:𝐶𝐵 ∧ ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶𝐵) ∧ (𝐶 ∩ (𝐴𝐶)) = ∅) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})):(𝐶 ∪ (𝐴𝐶))⟶𝐵)
8073, 76, 78, 79syl21anc 836 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})):(𝐶 ∪ (𝐴𝐶))⟶𝐵)
81 simprr 772 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
8281eqcomd 2827 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) = 𝑓)
8321ad2antrr 725 . . . . . . . . 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 6475 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍})):(𝐶 ∪ (𝐴𝐶))⟶𝐵𝑓:𝐴𝐵))
8680, 85mpbid 235 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑓:𝐴𝐵)
87 elmapg 8394 . . . . . . 7 ((𝐵𝑊𝐴𝑉) → (𝑓 ∈ (𝐵m 𝐴) ↔ 𝑓:𝐴𝐵))
8887biimpar 481 . . . . . 6 (((𝐵𝑊𝐴𝑉) ∧ 𝑓:𝐴𝐵) → 𝑓 ∈ (𝐵m 𝐴))
8970, 71, 86, 88syl21anc 836 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑓 ∈ (𝐵m 𝐴))
9071, 74, 86, 47syl3anc 1368 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓 supp 𝑍) = (𝑓 “ (𝐵 ∖ {𝑍})))
9181adantr 484 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
9291fveq1d 6645 . . . . . . . 8 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (𝑓𝑥) = ((𝑔 ∪ ((𝐴𝐶) × {𝑍}))‘𝑥))
9373adantr 484 . . . . . . . . . 10 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑔:𝐶𝐵)
9493ffnd 6488 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑔 Fn 𝐶)
95 fconstg 6539 . . . . . . . . . . 11 (𝑍𝐵 → ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶{𝑍})
9695ad3antlr 730 . . . . . . . . . 10 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶{𝑍})
9796ffnd 6488 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → ((𝐴𝐶) × {𝑍}) Fn (𝐴𝐶))
9877a1i 11 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (𝐶 ∩ (𝐴𝐶)) = ∅)
99 simpr 488 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑥 ∈ (𝐴𝐶))
100 fvun2 6728 . . . . . . . . 9 ((𝑔 Fn 𝐶 ∧ ((𝐴𝐶) × {𝑍}) Fn (𝐴𝐶) ∧ ((𝐶 ∩ (𝐴𝐶)) = ∅ ∧ 𝑥 ∈ (𝐴𝐶))) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍}))‘𝑥) = (((𝐴𝐶) × {𝑍})‘𝑥))
10194, 97, 98, 99, 100syl112anc 1371 . . . . . . . 8 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍}))‘𝑥) = (((𝐴𝐶) × {𝑍})‘𝑥))
102 fvconst 6899 . . . . . . . . 9 ((((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶{𝑍} ∧ 𝑥 ∈ (𝐴𝐶)) → (((𝐴𝐶) × {𝑍})‘𝑥) = 𝑍)
10396, 99, 102syl2anc 587 . . . . . . . 8 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (((𝐴𝐶) × {𝑍})‘𝑥) = 𝑍)
10492, 101, 1033eqtrd 2860 . . . . . . 7 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (𝑓𝑥) = 𝑍)
10586, 104suppss 7835 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓 supp 𝑍) ⊆ 𝐶)
10690, 105eqsstrrd 3982 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶)
10781reseq1d 5825 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓𝐶) = ((𝑔 ∪ ((𝐴𝐶) × {𝑍})) ↾ 𝐶))
108 res0 5830 . . . . . . . . . 10 (((𝐴𝐶) × {𝑍}) ↾ ∅) = ∅
109 res0 5830 . . . . . . . . . 10 (𝑔 ↾ ∅) = ∅
110108, 109eqtr4i 2847 . . . . . . . . 9 (((𝐴𝐶) × {𝑍}) ↾ ∅) = (𝑔 ↾ ∅)
11177reseq2i 5823 . . . . . . . . 9 (((𝐴𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴𝐶))) = (((𝐴𝐶) × {𝑍}) ↾ ∅)
11277reseq2i 5823 . . . . . . . . 9 (𝑔 ↾ (𝐶 ∩ (𝐴𝐶))) = (𝑔 ↾ ∅)
113110, 111, 1123eqtr4ri 2855 . . . . . . . 8 (𝑔 ↾ (𝐶 ∩ (𝐴𝐶))) = (((𝐴𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴𝐶)))
114113a1i 11 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑔 ↾ (𝐶 ∩ (𝐴𝐶))) = (((𝐴𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴𝐶))))
115 fresaunres1 6524 . . . . . . 7 ((𝑔:𝐶𝐵 ∧ ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶𝐵 ∧ (𝑔 ↾ (𝐶 ∩ (𝐴𝐶))) = (((𝐴𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴𝐶)))) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍})) ↾ 𝐶) = 𝑔)
11673, 76, 114, 115syl3anc 1368 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍})) ↾ 𝐶) = 𝑔)
117107, 116eqtr2d 2857 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑔 = (𝑓𝐶))
11889, 106, 117jca31 518 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶)))
11969, 118impbida 800 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → (((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶)) ↔ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))))
12016, 119syl5bb 286 . 2 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → ((𝑓𝑋𝑔 = (𝑓𝐶)) ↔ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))))
1211, 3, 13, 120f1od 7372 1 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → 𝐹:𝑋1-1-onto→(𝐵m 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  {crab 3130  Vcvv 3471   ∖ cdif 3907   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  ∅c0 4266  {csn 4540   ↦ cmpt 5119   × cxp 5526  ◡ccnv 5527   ↾ cres 5530   “ cima 5531   Fn wfn 6323  ⟶wf 6324  –1-1-onto→wf1o 6327  ‘cfv 6328  (class class class)co 7130   supp csupp 7805   ↑m cmap 8381 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-supp 7806  df-map 8383 This theorem is referenced by:  eulerpartgbij  31638
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