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Mirrors > Home > MPE Home > Th. List > fvmptdv | Structured version Visualization version GIF version |
Description: Alternate deduction version of fvmpt 6763, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) |
Ref | Expression |
---|---|
fvmptd2f.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
fvmptd2f.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) |
fvmptd2f.3 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓)) |
Ref | Expression |
---|---|
fvmptdv | ⊢ (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 | nfcv 2977 | . 2 ⊢ Ⅎ𝑥𝐹 | |
5 | nfv 1911 | . 2 ⊢ Ⅎ𝑥𝜓 | |
6 | 1, 2, 3, 4, 5 | fvmptd2f 6779 | 1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ↦ cmpt 5139 ‘cfv 6350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fv 6358 |
This theorem is referenced by: (None) |
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