Theorem cbvfo 7298

Description: Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Hypothesis

Ref	Expression
cbvfo.1	⊢ ((𝐹‘𝑥) = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvfo	⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐹,𝑦   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐵(𝑥)

Proof of Theorem cbvfo

Step	Hyp	Ref	Expression
1		fofn 6813	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
2		cbvfo.1	. . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝜑 ↔ 𝜓))
3	2	bicomd 222	. . . . 5 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝜓 ↔ 𝜑))
4	3	eqcoms 2736	. . . 4 ⊢ (𝑦 = (𝐹‘𝑥) → (𝜓 ↔ 𝜑))
5	4	ralrn 7098	. . 3 ⊢ (𝐹 Fn 𝐴 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜑))
6	1, 5	syl 17	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜑))
7		forn 6814	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
8	7	raleqdv 3322	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜓))
9	6, 8	bitr3d 281	1 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1534  ∀wral 3058  ran crn 5679   Fn wfn 6543  –onto→wfo 6546  ‘cfv 6548

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 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fo 6554  df-fv 6556

This theorem is referenced by:  cbvexfo  7299  cocan2  7301  f1oweALT  7976  supisolem  9496  qtopeu  23619  deg1leb  26030  dchrelbas4  27175  cnpconn  34840  cocanfo  37192  aks6d1c1p5  41583