Theorem fvelrnbf 45042

Description: A version of fvelrnb 6939 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
fvelrnbf.1	⊢ Ⅎ𝑥𝐴
fvelrnbf.2	⊢ Ⅎ𝑥𝐵
fvelrnbf.3	⊢ Ⅎ𝑥𝐹

Assertion

Ref	Expression
fvelrnbf	⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))

Proof of Theorem fvelrnbf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvelrnb 6939	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵))
2		nfcv 2898	. . 3 ⊢ Ⅎ𝑦𝐴
3		fvelrnbf.1	. . 3 ⊢ Ⅎ𝑥𝐴
4		fvelrnbf.3	. . . . 5 ⊢ Ⅎ𝑥𝐹
5		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	4, 5	nffv 6886	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
7		fvelrnbf.2	. . . 4 ⊢ Ⅎ𝑥𝐵
8	6, 7	nfeq 2912	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) = 𝐵
9		nfv 1914	. . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) = 𝐵
10		fveqeq2 6885	. . 3 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) = 𝐵 ↔ (𝐹‘𝑥) = 𝐵))
11	2, 3, 8, 9, 10	cbvrexfw 3285	. 2 ⊢ (∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵)
12	1, 11	bitrdi 287	1 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2108  Ⅎwnfc 2883  ∃wrex 3060  ran crn 5655   Fn wfn 6526  ‘cfv 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fn 6534  df-fv 6539

This theorem is referenced by:  refsumcn  45054  stoweidlem29  46058