Theorem funfvdm2f 5623

Description: The value of a function. Version of funfvdm2 5622 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 5622	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ {𝑤 ∣ 𝐴𝐹𝑤})
2		funfvdm2f.1	. . . . 5 ⊢ Ⅎ𝑦𝐴
3		funfvdm2f.2	. . . . 5 ⊢ Ⅎ𝑦𝐹
4		nfcv 2336	. . . . 5 ⊢ Ⅎ𝑦𝑤
5	2, 3, 4	nfbr 4076	. . . 4 ⊢ Ⅎ𝑦 𝐴𝐹𝑤
6		nfv 1539	. . . 4 ⊢ Ⅎ𝑤 𝐴𝐹𝑦
7		breq2 4034	. . . 4 ⊢ (𝑤 = 𝑦 → (𝐴𝐹𝑤 ↔ 𝐴𝐹𝑦))
8	5, 6, 7	cbvab 2317	. . 3 ⊢ {𝑤 ∣ 𝐴𝐹𝑤} = {𝑦 ∣ 𝐴𝐹𝑦}
9	8	unieqi 3846	. 2 ⊢ ∪ {𝑤 ∣ 𝐴𝐹𝑤} = ∪ {𝑦 ∣ 𝐴𝐹𝑦}
10	1, 9	eqtrdi 2242	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  {cab 2179  Ⅎwnfc 2323  ∪ cuni 3836   class class class wbr 4030  dom cdm 4660  Fun wfun 5249  ‘cfv 5255

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263

This theorem is referenced by: (None)