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Mirrors > Home > MPE Home > Th. List > Mathboxes > setrec2v | Structured version Visualization version GIF version |
Description: Version of setrec2 44105 with a disjoint variable condition instead of a non-freeness hypothesis. (Contributed by Emmett Weisz, 6-Mar-2021.) |
Ref | Expression |
---|---|
setrec2.b | ⊢ 𝐵 = setrecs(𝐹) |
setrec2.c | ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) |
Ref | Expression |
---|---|
setrec2v | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2926 | . 2 ⊢ Ⅎ𝑎𝐹 | |
2 | setrec2.b | . 2 ⊢ 𝐵 = setrecs(𝐹) | |
3 | setrec2.c | . 2 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) | |
4 | 1, 2, 3 | setrec2 44105 | 1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1505 = wceq 1507 ⊆ wss 3825 ‘cfv 6182 setrecscsetrecs 44093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3678 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fv 6190 df-setrecs 44094 |
This theorem is referenced by: setis 44107 elsetrecslem 44108 setrecsss 44110 setrecsres 44111 0setrec 44113 onsetrec 44117 |
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