Theorem nfimad 6070

Description: Deduction version of bound-variable hypothesis builder nfima 6069. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
nfimad.2	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfimad.3	⊢ (𝜑 → Ⅎ𝑥𝐵)

Assertion

Ref	Expression
nfimad	⊢ (𝜑 → Ⅎ𝑥(𝐴 “ 𝐵))

Proof of Theorem nfimad

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfaba1 2933	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}
2		nfaba1 2933	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}
3	1, 2	nfima 6069	. 2 ⊢ Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵})
4		nfimad.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5		nfimad.3	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐵)
6		nfnfc1 2928	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
7		nfnfc1 2928	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐵
8	6, 7	nfan 1933	. . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵)
9		abidnf 3662	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
10	9	imaeq1d 6056	. . . . 5 ⊢ (Ⅎ𝑥𝐴 → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}))
11		abidnf 3662	. . . . . 6 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵)
12	11	imaeq2d 6057	. . . . 5 ⊢ (Ⅎ𝑥𝐵 → (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ 𝐵))
13	10, 12	sylan9eq 2818	. . . 4 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ 𝐵))
14	8, 13	nfceqdf 2921	. . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) ↔ Ⅎ𝑥(𝐴 “ 𝐵)))
15	4, 5, 14	syl2anc 596	. 2 ⊢ (𝜑 → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) ↔ Ⅎ𝑥(𝐴 “ 𝐵)))
16	3, 15	mpbii 236	1 ⊢ (𝜑 → Ⅎ𝑥(𝐴 “ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   ∈ wcel 2146  {cab 2741  Ⅎwnfc 2910   “ cima 5657

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667

This theorem is used by: (None)