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Theorem cnvf1olem 8115

Description: Lemma for cnvf1o 8116. (Contributed by Mario Carneiro, 27-Apr-2014.)

**Assertion**
Ref	Expression
cnvf1olem	⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → (𝐶 ∈ ^◡𝐴 ∧ 𝐵 = ∪ ^◡{𝐶}))

**Proof of Theorem cnvf1olem**
Step	Hyp	Ref	Expression
1		simprr 771	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐶 = ∪ ^◡{𝐵})
2		1st2nd 8044	. . . . . . . . 9 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
3	2	adantrr 715	. . . . . . . 8 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
4	3	sneqd 4642	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → {𝐵} = {⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩})
5	4	cnveqd 5878	. . . . . 6 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ^◡{𝐵} = ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩})
6	5	unieqd 4922	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ∪ ^◡{𝐵} = ∪ ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩})
7	1, 6	eqtrd 2765	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐶 = ∪ ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩})
8		opswap 6235	. . . 4 ⊢ ∪ ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩} = ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩
9	7, 8	eqtrdi 2781	. . 3 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐶 = ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩)
10		simprl 769	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐵 ∈ 𝐴)
11	3, 10	eqeltrrd 2826	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ 𝐴)
12		fvex 6909	. . . . 5 ⊢ (2^nd ‘𝐵) ∈ V
13		fvex 6909	. . . . 5 ⊢ (1^st ‘𝐵) ∈ V
14	12, 13	opelcnv 5884	. . . 4 ⊢ (⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩ ∈ ^◡𝐴 ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ 𝐴)
15	11, 14	sylibr 233	. . 3 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩ ∈ ^◡𝐴)
16	9, 15	eqeltrd 2825	. 2 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐶 ∈ ^◡𝐴)
17		opswap 6235	. . . 4 ⊢ ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩} = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩
18	17	eqcomi 2734	. . 3 ⊢ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ = ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩}
19	9	sneqd 4642	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → {𝐶} = {⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
20	19	cnveqd 5878	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ^◡{𝐶} = ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
21	20	unieqd 4922	. . 3 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ∪ ^◡{𝐶} = ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
22	18, 3, 21	3eqtr4a 2791	. 2 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐵 = ∪ ^◡{𝐶})
23	16, 22	jca 510	1 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → (𝐶 ∈ ^◡𝐴 ∧ 𝐵 = ∪ ^◡{𝐶}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {csn 4630 ⟨cop 4636 ∪ cuni 4909 ^◡ccnv 5677 Rel wrel 5683 ‘cfv 6549 1^st c1st 7992 2^nd c2nd 7993

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6501 df-fun 6551 df-fv 6557 df-1st 7994 df-2nd 7995

This theorem is referenced by: cnvf1o 8116 fcnvgreu 32540 gsumhashmul 32860

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