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Mirrors > Home > MPE Home > Th. List > strfv2 | Structured version Visualization version GIF version |
Description: A variation on strfv 16533 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 16531 | 1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 〈cop 4575 ◡ccnv 5556 Fun wfun 6351 ‘cfv 6357 ndxcnx 16482 Slot cslot 16484 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-res 5569 df-iota 6316 df-fun 6359 df-fv 6365 df-slot 16489 |
This theorem is referenced by: strfv 16533 |
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