Theorem fvelrnbf 45019

Description: A version of fvelrnb 6924 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 6924	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵))
2		nfcv 2892	. . 3 ⊢ Ⅎ𝑦𝐴
3		fvelrnbf.1	. . 3 ⊢ Ⅎ𝑥𝐴
4		fvelrnbf.3	. . . . 5 ⊢ Ⅎ𝑥𝐹
5		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	4, 5	nffv 6871	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
7		fvelrnbf.2	. . . 4 ⊢ Ⅎ𝑥𝐵
8	6, 7	nfeq 2906	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) = 𝐵
9		nfv 1914	. . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) = 𝐵
10		fveqeq2 6870	. . 3 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) = 𝐵 ↔ (𝐹‘𝑥) = 𝐵))
11	2, 3, 8, 9, 10	cbvrexfw 3281	. 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 2109  Ⅎwnfc 2877  ∃wrex 3054  ran crn 5642   Fn wfn 6509  ‘cfv 6514

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2066  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-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fn 6517  df-fv 6522

This theorem is referenced by:  refsumcn  45031  stoweidlem29  46034