Theorem f1ocnvfv3 7336

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 7214	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (◡𝐹‘𝐶) ∈ 𝐴)
2		f1ocnvfvb 7208	. . . . . 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 652	. . . 4 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ (◡𝐹‘𝐶) = 𝑥))
5		eqcom 2737	. . . 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 7327	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (℩𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶) = (◡𝐹‘𝐶))
8	7	eqcomd 2736	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 1541   ∈ wcel 2110  ^◡ccnv 5613  –1-1-onto→wf1o 6476  ‘cfv 6477  ℩crio 7297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298

This theorem is referenced by: (None)