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Mirrors > Home > MPE Home > Th. List > Mathboxes > ecexALTV | Structured version Visualization version GIF version |
Description: Existence of a coset, like ecexg 8707 but with a weaker antecedent: only the restricion of 𝑅 by the singleton of 𝐴 needs to be a set, not 𝑅 itself, see e.g. eccnvepex 37204. (Contributed by Peter Mazsa, 22-Feb-2023.) |
Ref | Expression |
---|---|
ecexALTV | ⊢ ((𝑅 ↾ {𝐴}) ∈ 𝑉 → [𝐴]𝑅 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ec 8705 | . 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | |
2 | snex 5432 | . . 3 ⊢ {𝐴} ∈ V | |
3 | imaexALTV 37199 | . . . 4 ⊢ ((𝑅 ∈ V ∨ ((𝑅 ↾ {𝐴}) ∈ 𝑉 ∧ {𝐴} ∈ V)) → (𝑅 “ {𝐴}) ∈ V) | |
4 | 3 | olcs 875 | . . 3 ⊢ (((𝑅 ↾ {𝐴}) ∈ 𝑉 ∧ {𝐴} ∈ V) → (𝑅 “ {𝐴}) ∈ V) |
5 | 2, 4 | mpan2 690 | . 2 ⊢ ((𝑅 ↾ {𝐴}) ∈ 𝑉 → (𝑅 “ {𝐴}) ∈ V) |
6 | 1, 5 | eqeltrid 2838 | 1 ⊢ ((𝑅 ↾ {𝐴}) ∈ 𝑉 → [𝐴]𝑅 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 Vcvv 3475 {csn 4629 ↾ cres 5679 “ cima 5680 [cec 8701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ec 8705 df-qs 8709 |
This theorem is referenced by: eccnvepex 37204 |
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