Theorem fvelrnbf 45378

Description: A version of fvelrnb 6902 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 6902	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵))
2		nfcv 2899	. . 3 ⊢ Ⅎ𝑦𝐴
3		fvelrnbf.1	. . 3 ⊢ Ⅎ𝑥𝐴
4		fvelrnbf.3	. . . . 5 ⊢ Ⅎ𝑥𝐹
5		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	4, 5	nffv 6852	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
7		fvelrnbf.2	. . . 4 ⊢ Ⅎ𝑥𝐵
8	6, 7	nfeq 2913	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) = 𝐵
9		nfv 1916	. . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) = 𝐵
10		fveqeq2 6851	. . 3 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) = 𝐵 ↔ (𝐹‘𝑥) = 𝐵))
11	2, 3, 8, 9, 10	cbvrexfw 3279	. 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 1542   ∈ wcel 2114  Ⅎwnfc 2884  ∃wrex 3062  ran crn 5633   Fn wfn 6495  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-fv 6508

This theorem is referenced by:  refsumcn  45390  stoweidlem29  46387