Theorem fvelrnbf 45852

Description: A version of fvelrnb 6938 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 6938	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵))
2		nfcv 2922	. . 3 ⊢ Ⅎ𝑦𝐴
3		fvelrnbf.1	. . 3 ⊢ Ⅎ𝑥𝐴
4		fvelrnbf.3	. . . . 5 ⊢ Ⅎ𝑥𝐹
5		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	4, 5	nffv 6888	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
7		fvelrnbf.2	. . . 4 ⊢ Ⅎ𝑥𝐵
8	6, 7	nfeq 2935	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) = 𝐵
9		nfv 1947	. . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) = 𝐵
10		fveqeq2 6887	. . 3 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) = 𝐵 ↔ (𝐹‘𝑥) = 𝐵))
11	2, 3, 8, 9, 10	cbvrexfw 3303	. 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 2145  Ⅎwnfc 2907  ∃wrex 3086  ran crn 5656   Fn wfn 6528  ‘cfv 6533

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541

This theorem is used by:  refsumcn  45864  stoweidlem29  46857