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Mirrors > Home > ILE Home > Th. List > nfimad | GIF version |
Description: Deduction version of bound-variable hypothesis builder nfima 4845. (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 2259 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} | |
2 | nfaba1 2259 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} | |
3 | 1, 2 | nfima 4845 | . 2 ⊢ Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) |
4 | nfimad.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | nfimad.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
6 | nfnfc1 2256 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
7 | nfnfc1 2256 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐵 | |
8 | 6, 7 | nfan 1525 | . . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) |
9 | abidnf 2819 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | |
10 | 9 | imaeq1d 4836 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵})) |
11 | abidnf 2819 | . . . . . 6 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵) | |
12 | 11 | imaeq2d 4837 | . . . . 5 ⊢ (Ⅎ𝑥𝐵 → (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ 𝐵)) |
13 | 10, 12 | sylan9eq 2165 | . . . 4 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ 𝐵)) |
14 | 8, 13 | nfceqdf 2252 | . . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) ↔ Ⅎ𝑥(𝐴 “ 𝐵))) |
15 | 4, 5, 14 | syl2anc 406 | . 2 ⊢ (𝜑 → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) ↔ Ⅎ𝑥(𝐴 “ 𝐵))) |
16 | 3, 15 | mpbii 147 | 1 ⊢ (𝜑 → Ⅎ𝑥(𝐴 “ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1310 ∈ wcel 1461 {cab 2099 Ⅎwnfc 2240 “ cima 4500 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-rab 2397 df-v 2657 df-un 3039 df-in 3041 df-ss 3048 df-sn 3497 df-pr 3498 df-op 3500 df-br 3894 df-opab 3948 df-xp 4503 df-cnv 4505 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 |
This theorem is referenced by: (None) |
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