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Mirrors > Home > MPE Home > Th. List > nfimad | Structured version Visualization version GIF version |
Description: Deduction version of bound-variable hypothesis builder nfima 5772. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
nfimad.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfimad.3 | ⊢ (𝜑 → Ⅎ𝑥𝐵) |
Ref | Expression |
---|---|
nfimad | ⊢ (𝜑 → Ⅎ𝑥(𝐴 “ 𝐵)) |
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 5772 | . 2 ⊢ Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) |
4 | nfimad.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | nfimad.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
6 | nfnfc1 2929 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
7 | nfnfc1 2929 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐵 | |
8 | 6, 7 | nfan 1862 | . . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) |
9 | abidnf 3602 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | |
10 | 9 | imaeq1d 5763 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵})) |
11 | abidnf 3602 | . . . . . 6 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵) | |
12 | 11 | imaeq2d 5764 | . . . . 5 ⊢ (Ⅎ𝑥𝐵 → (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ 𝐵)) |
13 | 10, 12 | sylan9eq 2828 | . . . 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 576 | . 2 ⊢ (𝜑 → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) ↔ Ⅎ𝑥(𝐴 “ 𝐵))) |
16 | 3, 15 | mpbii 225 | 1 ⊢ (𝜑 → Ⅎ𝑥(𝐴 “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∀wal 1505 ∈ wcel 2048 {cab 2753 Ⅎwnfc 2910 “ cima 5403 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-rab 3091 df-v 3411 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4924 df-opab 4986 df-xp 5406 df-cnv 5408 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 |
This theorem is referenced by: (None) |
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