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Theorem resf1o 32641
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 6026 . . 3 (𝑓𝑋 → (𝑓𝐶) ∈ V)
32adantl 480 . 2 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ 𝑓𝑋) → (𝑓𝐶) ∈ V)
4 simpr 483 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑔 ∈ (𝐵m 𝐶)) → 𝑔 ∈ (𝐵m 𝐶))
5 difexg 5324 . . . . . . 7 (𝐴𝑉 → (𝐴𝐶) ∈ V)
653ad2ant1 1130 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝐴) → (𝐴𝐶) ∈ V)
7 snex 5427 . . . . . 6 {𝑍} ∈ V
8 xpexg 7747 . . . . . 6 (((𝐴𝐶) ∈ V ∧ {𝑍} ∈ V) → ((𝐴𝐶) × {𝑍}) ∈ V)
96, 7, 8sylancl 584 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝐴) → ((𝐴𝐶) × {𝑍}) ∈ V)
109adantr 479 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑔 ∈ (𝐵m 𝐶)) → ((𝐴𝐶) × {𝑍}) ∈ V)
11 unexg 7746 . . . 4 ((𝑔 ∈ (𝐵m 𝐶) ∧ ((𝐴𝐶) × {𝑍}) ∈ V) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) ∈ V)
124, 10, 11syl2anc 582 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑔 ∈ (𝐵m 𝐶)) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) ∈ V)
1312adantlr 713 . 2 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ 𝑔 ∈ (𝐵m 𝐶)) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) ∈ V)
14 resf1o.1 . . . . 5 𝑋 = {𝑓 ∈ (𝐵m 𝐴) ∣ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶}
1514reqabi 3442 . . . 4 (𝑓𝑋 ↔ (𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶))
1615anbi1i 622 . . 3 ((𝑓𝑋𝑔 = (𝑓𝐶)) ↔ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶)))
17 simprr 771 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑔 = (𝑓𝐶))
18 simprll 777 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 ∈ (𝐵m 𝐴))
19 elmapi 8867 . . . . . . . . 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 724 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝐶𝐴)
2320, 22fssresd 6758 . . . . . . 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 5319 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊𝐶𝐴) → 𝐶 ∈ V)
27 elmapg 8857 . . . . . . . . 9 ((𝐵𝑊𝐶 ∈ V) → ((𝑓𝐶) ∈ (𝐵m 𝐶) ↔ (𝑓𝐶):𝐶𝐵))
2824, 26, 27syl2anc 582 . . . . . . . 8 ((𝐴𝑉𝐵𝑊𝐶𝐴) → ((𝑓𝐶) ∈ (𝐵m 𝐶) ↔ (𝑓𝐶):𝐶𝐵))
2928ad2antrr 724 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → ((𝑓𝐶) ∈ (𝐵m 𝐶) ↔ (𝑓𝐶):𝐶𝐵))
3023, 29mpbird 256 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓𝐶) ∈ (𝐵m 𝐶))
3117, 30eqeltrd 2826 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑔 ∈ (𝐵m 𝐶))
32 undif 4476 . . . . . . . . . . 11 (𝐶𝐴 ↔ (𝐶 ∪ (𝐴𝐶)) = 𝐴)
3332biimpi 215 . . . . . . . . . 10 (𝐶𝐴 → (𝐶 ∪ (𝐴𝐶)) = 𝐴)
3433reseq2d 5979 . . . . . . . . 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 6717 . . . . . . . . 9 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
37 fnresdm 6669 . . . . . . . . 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 2767 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 = (𝑓 ↾ (𝐶 ∪ (𝐴𝐶))))
40 resundi 5993 . . . . . . 7 (𝑓 ↾ (𝐶 ∪ (𝐴𝐶))) = ((𝑓𝐶) ∪ (𝑓 ↾ (𝐴𝐶)))
4139, 40eqtrdi 2782 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 = ((𝑓𝐶) ∪ (𝑓 ↾ (𝐴𝐶))))
4217eqcomd 2732 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓𝐶) = 𝑔)
43 simprlr 778 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶)
4425ad2antrr 724 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝐴𝑉)
45 simplr 767 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑍𝐵)
46 eqid 2726 . . . . . . . . . . 11 (𝐵 ∖ {𝑍}) = (𝐵 ∖ {𝑍})
4746ffs2 32639 . . . . . . . . . 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 4213 . . . . . . . . . . 11 (𝐶𝐴 ↔ (𝐴𝐶) = 𝐶)
5049biimpi 215 . . . . . . . . . 10 (𝐶𝐴 → (𝐴𝐶) = 𝐶)
5122, 50syl 17 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝐴𝐶) = 𝐶)
5243, 48, 513sstr4d 4026 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 supp 𝑍) ⊆ (𝐴𝐶))
53 simpl 481 . . . . . . . . . . . 12 ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑍𝐵) → 𝑓 ∈ (𝐵m 𝐴))
5453, 19, 363syl 18 . . . . . . . . . . 11 ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑍𝐵) → 𝑓 Fn 𝐴)
55 inundif 4473 . . . . . . . . . . . 12 ((𝐴𝐶) ∪ (𝐴𝐶)) = 𝐴
5655fneq2i 6647 . . . . . . . . . . 11 (𝑓 Fn ((𝐴𝐶) ∪ (𝐴𝐶)) ↔ 𝑓 Fn 𝐴)
5754, 56sylibr 233 . . . . . . . . . 10 ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑍𝐵) → 𝑓 Fn ((𝐴𝐶) ∪ (𝐴𝐶)))
58 vex 3466 . . . . . . . . . . 11 𝑓 ∈ V
5958a1i 11 . . . . . . . . . 10 ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑍𝐵) → 𝑓 ∈ V)
60 simpr 483 . . . . . . . . . 10 ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑍𝐵) → 𝑍𝐵)
61 inindif 32441 . . . . . . . . . . 11 ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅
6261a1i 11 . . . . . . . . . 10 ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑍𝐵) → ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅)
63 fnsuppres 8194 . . . . . . . . . 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 582 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → ((𝑓 supp 𝑍) ⊆ (𝐴𝐶) ↔ (𝑓 ↾ (𝐴𝐶)) = ((𝐴𝐶) × {𝑍})))
6652, 65mpbid 231 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 ↾ (𝐴𝐶)) = ((𝐴𝐶) × {𝑍}))
6742, 66uneq12d 4161 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → ((𝑓𝐶) ∪ (𝑓 ↾ (𝐴𝐶))) = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
6841, 67eqtrd 2766 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
6931, 68jca 510 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍}))))
7024ad2antrr 724 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝐵𝑊)
7125ad2antrr 724 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝐴𝑉)
72 elmapi 8867 . . . . . . . . 9 (𝑔 ∈ (𝐵m 𝐶) → 𝑔:𝐶𝐵)
7372ad2antrl 726 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑔:𝐶𝐵)
74 simplr 767 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑍𝐵)
75 fconst6g 6780 . . . . . . . . 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 4466 . . . . . . . . 9 (𝐶 ∩ (𝐴𝐶)) = ∅
7877a1i 11 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝐶 ∩ (𝐴𝐶)) = ∅)
79 fun2 6754 . . . . . . . 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 771 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
8281eqcomd 2732 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) = 𝑓)
8321ad2antrr 724 . . . . . . . . 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 6705 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍})):(𝐶 ∪ (𝐴𝐶))⟶𝐵𝑓:𝐴𝐵))
8680, 85mpbid 231 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑓:𝐴𝐵)
87 elmapg 8857 . . . . . . 7 ((𝐵𝑊𝐴𝑉) → (𝑓 ∈ (𝐵m 𝐴) ↔ 𝑓:𝐴𝐵))
8887biimpar 476 . . . . . 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 479 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
9291fveq1d 6892 . . . . . . . 8 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (𝑓𝑥) = ((𝑔 ∪ ((𝐴𝐶) × {𝑍}))‘𝑥))
9373adantr 479 . . . . . . . . . 10 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑔:𝐶𝐵)
9493ffnd 6718 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑔 Fn 𝐶)
95 fconstg 6778 . . . . . . . . . . 11 (𝑍𝐵 → ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶{𝑍})
9695ad3antlr 729 . . . . . . . . . 10 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶{𝑍})
9796ffnd 6718 . . . . . . . . 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 483 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑥 ∈ (𝐴𝐶))
100 fvun2 6983 . . . . . . . . 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 7167 . . . . . . . . 9 ((((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶{𝑍} ∧ 𝑥 ∈ (𝐴𝐶)) → (((𝐴𝐶) × {𝑍})‘𝑥) = 𝑍)
10396, 99, 102syl2anc 582 . . . . . . . 8 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (((𝐴𝐶) × {𝑍})‘𝑥) = 𝑍)
10492, 101, 1033eqtrd 2770 . . . . . . 7 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (𝑓𝑥) = 𝑍)
10586, 104suppss 8197 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓 supp 𝑍) ⊆ 𝐶)
10690, 105eqsstrrd 4018 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶)
10781reseq1d 5978 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓𝐶) = ((𝑔 ∪ ((𝐴𝐶) × {𝑍})) ↾ 𝐶))
108 res0 5983 . . . . . . . . . 10 (((𝐴𝐶) × {𝑍}) ↾ ∅) = ∅
109 res0 5983 . . . . . . . . . 10 (𝑔 ↾ ∅) = ∅
110108, 109eqtr4i 2757 . . . . . . . . 9 (((𝐴𝐶) × {𝑍}) ↾ ∅) = (𝑔 ↾ ∅)
11177reseq2i 5976 . . . . . . . . 9 (((𝐴𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴𝐶))) = (((𝐴𝐶) × {𝑍}) ↾ ∅)
11277reseq2i 5976 . . . . . . . . 9 (𝑔 ↾ (𝐶 ∩ (𝐴𝐶))) = (𝑔 ↾ ∅)
113110, 111, 1123eqtr4ri 2765 . . . . . . . 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 6764 . . . . . . 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 2767 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑔 = (𝑓𝐶))
11889, 106, 117jca31 513 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → ((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶)))
11969, 118impbida 799 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → (((𝑓 ∈ (𝐵m 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶)) ↔ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))))
12016, 119bitrid 282 . 2 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → ((𝑓𝑋𝑔 = (𝑓𝐶)) ↔ (𝑔 ∈ (𝐵m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))))
1211, 3, 13, 120f1od 7667 1 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → 𝐹:𝑋1-1-onto→(𝐵m 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  {crab 3419  Vcvv 3462  cdif 3943  cun 3944  cin 3945  wss 3946  c0 4322  {csn 4623  cmpt 5226   × cxp 5670  ccnv 5671  cres 5674  cima 5675   Fn wfn 6538  wf 6539  1-1-ontowf1o 6542  cfv 6543  (class class class)co 7413   supp csupp 8163  m cmap 8844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7992  df-2nd 7993  df-supp 8164  df-map 8846
This theorem is referenced by:  eulerpartgbij  34216
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