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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 6085. (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 2912 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} | |
| 2 | nfaba1 2912 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} | |
| 3 | 1, 2 | nfima 6085 | . 2 ⊢ Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) | 
| 4 | nfimad.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
| 5 | nfimad.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
| 6 | nfnfc1 2907 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
| 7 | nfnfc1 2907 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐵 | |
| 8 | 6, 7 | nfan 1898 | . . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) | 
| 9 | abidnf 3707 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | |
| 10 | 9 | imaeq1d 6076 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵})) | 
| 11 | abidnf 3707 | . . . . . 6 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵) | |
| 12 | 11 | imaeq2d 6077 | . . . . 5 ⊢ (Ⅎ𝑥𝐵 → (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ 𝐵)) | 
| 13 | 10, 12 | sylan9eq 2796 | . . . 4 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ 𝐵)) | 
| 14 | 8, 13 | nfceqdf 2900 | . . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) ↔ Ⅎ𝑥(𝐴 “ 𝐵))) | 
| 15 | 4, 5, 14 | syl2anc 584 | . 2 ⊢ (𝜑 → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) ↔ Ⅎ𝑥(𝐴 “ 𝐵))) | 
| 16 | 3, 15 | mpbii 233 | 1 ⊢ (𝜑 → Ⅎ𝑥(𝐴 “ 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1537 ∈ wcel 2107 {cab 2713 Ⅎwnfc 2889 “ cima 5687 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 | 
| This theorem is referenced by: (None) | 
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