Theorem funfvdm2f 5657

Description: The value of a function. Version of funfvdm2 5656 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 1-Jan-2019.)

Hypotheses

Ref	Expression
funfvdm2f.1	⊢ Ⅎ𝑦𝐴
funfvdm2f.2	⊢ Ⅎ𝑦𝐹

Assertion

Ref	Expression
funfvdm2f	⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦})

Proof of Theorem funfvdm2f

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfvdm2 5656	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ {𝑤 ∣ 𝐴𝐹𝑤})
2		funfvdm2f.1	. . . . 5 ⊢ Ⅎ𝑦𝐴
3		funfvdm2f.2	. . . . 5 ⊢ Ⅎ𝑦𝐹
4		nfcv 2349	. . . . 5 ⊢ Ⅎ𝑦𝑤
5	2, 3, 4	nfbr 4098	. . . 4 ⊢ Ⅎ𝑦 𝐴𝐹𝑤
6		nfv 1552	. . . 4 ⊢ Ⅎ𝑤 𝐴𝐹𝑦
7		breq2 4055	. . . 4 ⊢ (𝑤 = 𝑦 → (𝐴𝐹𝑤 ↔ 𝐴𝐹𝑦))
8	5, 6, 7	cbvab 2330	. . 3 ⊢ {𝑤 ∣ 𝐴𝐹𝑤} = {𝑦 ∣ 𝐴𝐹𝑦}
9	8	unieqi 3866	. 2 ⊢ ∪ {𝑤 ∣ 𝐴𝐹𝑤} = ∪ {𝑦 ∣ 𝐴𝐹𝑦}
10	1, 9	eqtrdi 2255	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  {cab 2192  Ⅎwnfc 2336  ∪ cuni 3856   class class class wbr 4051  dom cdm 4683  Fun wfun 5274  ‘cfv 5280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-fv 5288

This theorem is referenced by: (None)