Theorem f1ocnvfvb 6763

Description: Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)

Assertion

Ref	Expression
f1ocnvfvb	⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐹‘𝐶) = 𝐷 ↔ (◡𝐹‘𝐷) = 𝐶))

Proof of Theorem f1ocnvfvb

Step	Hyp	Ref	Expression
1		f1ocnvfv 6762	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = 𝐶))
2	1	3adant3 1163	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = 𝐶))
3		fveq2 6411	. . . . 5 ⊢ (𝐶 = (◡𝐹‘𝐷) → (𝐹‘𝐶) = (𝐹‘(◡𝐹‘𝐷)))
4	3	eqcoms 2807	. . . 4 ⊢ ((◡𝐹‘𝐷) = 𝐶 → (𝐹‘𝐶) = (𝐹‘(◡𝐹‘𝐷)))
5		f1ocnvfv2 6761	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝐷)) = 𝐷)
6	5	eqeq2d 2809	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐷 ∈ 𝐵) → ((𝐹‘𝐶) = (𝐹‘(◡𝐹‘𝐷)) ↔ (𝐹‘𝐶) = 𝐷))
7	4, 6	syl5ib 236	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐷 ∈ 𝐵) → ((◡𝐹‘𝐷) = 𝐶 → (𝐹‘𝐶) = 𝐷))
8	7	3adant2 1162	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((◡𝐹‘𝐷) = 𝐶 → (𝐹‘𝐶) = 𝐷))
9	2, 8	impbid 204	1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐹‘𝐶) = 𝐷 ↔ (◡𝐹‘𝐷) = 𝐶))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157  ^◡ccnv 5311  –1-1-onto→wf1o 6100  ‘cfv 6101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109

This theorem is referenced by:  f1ofveu  6873  f1ocnvfv3  6874  1arith2  15965  f1omvdcnv  18176  f1omvdconj  18178  txhmeo  21935  iccpnfcnv  23071  dvcnvlem  24080  logeftb  24671  sqff1o  25260  bracnlnval  29498  cdlemg17h  36689  isomuspgrlem1  42497