Theorem fvelrnbf 46309

Description: A version of fvelrnb 6976 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 6976	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵))
2		nfcv 2926	. . 3 ⊢ Ⅎ𝑦𝐴
3		fvelrnbf.1	. . 3 ⊢ Ⅎ𝑥𝐴
4		fvelrnbf.3	. . . . 5 ⊢ Ⅎ𝑥𝐹
5		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	4, 5	nffv 6926	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
7		fvelrnbf.2	. . . 4 ⊢ Ⅎ𝑥𝐵
8	6, 7	nfeq 2939	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) = 𝐵
9		nfv 1949	. . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) = 𝐵
10		fveqeq2 6925	. . 3 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) = 𝐵 ↔ (𝐹‘𝑥) = 𝐵))
11	2, 3, 8, 9, 10	cbvrexfw 3307	. 2 ⊢ (∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵)
12	1, 11	bitrdi 290	1 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  Ⅎwnfc 2911  ∃wrex 3090  ran crn 5659   Fn wfn 6561  ‘cfv 6566

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fn 6569  df-fv 6574

This theorem is used by:  refsumcn  46321  stoweidlem29  47312