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Mirrors > Home > MPE Home > Th. List > fvmptd2f | Structured version Visualization version GIF version |
Description: Alternate deduction version of fvmpt 7010, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) (Proof shortened by AV, 19-Jan-2022.) |
Ref | Expression |
---|---|
fvmptd2f.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
fvmptd2f.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) |
fvmptd2f.3 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓)) |
fvmptd2f.4 | ⊢ Ⅎ𝑥𝐹 |
fvmptd2f.5 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
fvmptd2f | ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptd2f.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
2 | fvmptd2f.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) | |
3 | fvmptd2f.3 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓)) | |
4 | fvmptd2f.4 | . 2 ⊢ Ⅎ𝑥𝐹 | |
5 | fvmptd2f.5 | . 2 ⊢ Ⅎ𝑥𝜓 | |
6 | nfv 1909 | . 2 ⊢ Ⅎ𝑥𝜑 | |
7 | 1, 2, 3, 4, 5, 6 | fvmptd3f 7025 | 1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2879 ↦ cmpt 5235 ‘cfv 6553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fv 6561 |
This theorem is referenced by: fvmptdv 7027 yonedalem4b 18275 |
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