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Mirrors > Home > MPE Home > Th. List > fvmptdf | Structured version Visualization version GIF version |
Description: Alternate deduction version of fvmpt 6761, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) (Proof shortened by AV, 19-Jan-2022.) |
Ref | Expression |
---|---|
fvmptdf.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
fvmptdf.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) |
fvmptdf.3 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓)) |
fvmptdf.4 | ⊢ Ⅎ𝑥𝐹 |
fvmptdf.5 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
fvmptdf | ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptdf.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
2 | fvmptdf.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) | |
3 | fvmptdf.3 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓)) | |
4 | fvmptdf.4 | . 2 ⊢ Ⅎ𝑥𝐹 | |
5 | fvmptdf.5 | . 2 ⊢ Ⅎ𝑥𝜓 | |
6 | nfv 1906 | . 2 ⊢ Ⅎ𝑥𝜑 | |
7 | 1, 2, 3, 4, 5, 6 | fvmptd3f 6775 | 1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 Ⅎwnfc 2958 ↦ cmpt 5137 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 |
This theorem is referenced by: fvmptdv 6777 yonedalem4b 17514 |
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