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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fmptdff | Structured version Visualization version GIF version | ||
| Description: A version of fmptd 7133 using bound-variable hypothesis instead of a distinct variable condition for 𝜑. (Contributed by Glauco Siliprandi, 5-Jan-2025.) | 
| Ref | Expression | 
|---|---|
| fmptdff.1 | ⊢ Ⅎ𝑥𝜑 | 
| fmptdff.2 | ⊢ Ⅎ𝑥𝐴 | 
| fmptdff.3 | ⊢ Ⅎ𝑥𝐶 | 
| fmptdff.4 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) | 
| fmptdff.5 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | 
| Ref | Expression | 
|---|---|
| fmptdff | ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fmptdff.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
| 2 | fmptdff.4 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) | |
| 3 | 1, 2 | ralrimia 3257 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) | 
| 4 | fmptdff.2 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
| 5 | fmptdff.3 | . . 3 ⊢ Ⅎ𝑥𝐶 | |
| 6 | fmptdff.5 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 7 | 4, 5, 6 | fmptff 45281 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶) | 
| 8 | 3, 7 | sylib 218 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 Ⅎwnf 1782 ∈ wcel 2107 Ⅎwnfc 2889 ∀wral 3060 ↦ cmpt 5224 ⟶wf 6556 | 
| 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 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| 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-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 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-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-fun 6562 df-fn 6563 df-f 6564 | 
| This theorem is referenced by: (None) | 
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