Theorem funfv2f 6963

Description: The value of a function. Version of funfv2 6962 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.)

Hypotheses

Ref	Expression
funfv2f.1	⊢ Ⅎ𝑦𝐴
funfv2f.2	⊢ Ⅎ𝑦𝐹

Assertion

Ref	Expression
funfv2f	⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦})

Proof of Theorem funfv2f

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfv2 6962	. 2 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑤 ∣ 𝐴𝐹𝑤})
2		funfv2f.1	. . . . 5 ⊢ Ⅎ𝑦𝐴
3		funfv2f.2	. . . . 5 ⊢ Ⅎ𝑦𝐹
4		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑦𝑤
5	2, 3, 4	nfbr 5185	. . . 4 ⊢ Ⅎ𝑦 𝐴𝐹𝑤
6		nfv 1917	. . . 4 ⊢ Ⅎ𝑤 𝐴𝐹𝑦
7		breq2 5142	. . . 4 ⊢ (𝑤 = 𝑦 → (𝐴𝐹𝑤 ↔ 𝐴𝐹𝑦))
8	5, 6, 7	cbvabw 2805	. . 3 ⊢ {𝑤 ∣ 𝐴𝐹𝑤} = {𝑦 ∣ 𝐴𝐹𝑦}
9	8	unieqi 4911	. 2 ⊢ ∪ {𝑤 ∣ 𝐴𝐹𝑤} = ∪ {𝑦 ∣ 𝐴𝐹𝑦}
10	1, 9	eqtrdi 2787	1 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦})

Syntax hints:   → wi 4   = wceq 1541  {cab 2708  Ⅎwnfc 2882  ∪ cuni 4898   class class class wbr 5138  Fun wfun 6523  ‘cfv 6529

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3430  df-v 3472  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6481  df-fun 6531  df-fn 6532  df-fv 6537

This theorem is referenced by: (None)