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Mirrors > Home > MPE Home > Th. List > Mathboxes > setrec2v | Structured version Visualization version GIF version |
Description: Version of setrec2 48926 with a disjoint variable condition instead of a nonfreeness 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 2903 | . 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 48926 | 1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 = wceq 1537 ⊆ wss 3963 ‘cfv 6563 setrecscsetrecs 48914 |
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-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
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-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 df-setrecs 48915 |
This theorem is referenced by: setis 48929 elsetrecslem 48930 setrecsss 48932 setrecsres 48933 0setrec 48935 onsetrec 48939 |
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