Theorem f1ocnvfv3 7443

Description: Value of the converse of a one-to-one onto function. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Assertion

Ref	Expression
f1ocnvfv3	⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (◡𝐹‘𝐶) = (℩𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹

Proof of Theorem f1ocnvfv3

Step	Hyp	Ref	Expression
1		f1ocnvdm 7321	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (◡𝐹‘𝐶) ∈ 𝐴)
2		f1ocnvfvb 7315	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝐹‘𝑥) = 𝐶 ↔ (◡𝐹‘𝐶) = 𝑥))
3	2	3expa 1118	. . . . 5 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝐶 ∈ 𝐵) → ((𝐹‘𝑥) = 𝐶 ↔ (◡𝐹‘𝐶) = 𝑥))
4	3	an32s 651	. . . 4 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ (◡𝐹‘𝐶) = 𝑥))
5		eqcom 2747	. . . 4 ⊢ (𝑥 = (◡𝐹‘𝐶) ↔ (◡𝐹‘𝐶) = 𝑥)
6	4, 5	bitr4di 289	. . 3 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ 𝑥 = (◡𝐹‘𝐶)))
7	1, 6	riota5 7434	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (℩𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶) = (◡𝐹‘𝐶))
8	7	eqcomd 2746	1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (◡𝐹‘𝐶) = (℩𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ^◡ccnv 5699  –1-1-onto→wf1o 6572  ‘cfv 6573  ℩crio 7403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404

This theorem is referenced by: (None)