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Mirrors > Home > MPE Home > Th. List > strfv2 | Structured version Visualization version GIF version |
Description: A variation on strfv 17238 to avoid asserting that 𝑆 itself is a function, which involves sethood of all the ordered pair components of 𝑆. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
strfv2.s | ⊢ 𝑆 ∈ V |
strfv2.f | ⊢ Fun ◡◡𝑆 |
strfv2.e | ⊢ 𝐸 = Slot (𝐸‘ndx) |
strfv2.n | ⊢ 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 |
Ref | Expression |
---|---|
strfv2 | ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strfv2.e | . 2 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
2 | strfv2.s | . . 3 ⊢ 𝑆 ∈ V | |
3 | 2 | a1i 11 | . 2 ⊢ (𝐶 ∈ 𝑉 → 𝑆 ∈ V) |
4 | strfv2.f | . . 3 ⊢ Fun ◡◡𝑆 | |
5 | 4 | a1i 11 | . 2 ⊢ (𝐶 ∈ 𝑉 → Fun ◡◡𝑆) |
6 | strfv2.n | . . 3 ⊢ 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 | |
7 | 6 | a1i 11 | . 2 ⊢ (𝐶 ∈ 𝑉 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) |
8 | id 22 | . 2 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ 𝑉) | |
9 | 1, 3, 5, 7, 8 | strfv2d 17236 | 1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 ◡ccnv 5688 Fun wfun 6557 ‘cfv 6563 Slot cslot 17215 ndxcnx 17227 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-res 5701 df-iota 6516 df-fun 6565 df-fv 6571 df-slot 17216 |
This theorem is referenced by: strfv 17238 |
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