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Theorem ecexALTV 37200
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.)
Assertion
Ref Expression
ecexALTV ((𝑅 ↾ {𝐴}) ∈ 𝑉 → [𝐴]𝑅 ∈ V)

Proof of Theorem ecexALTV
StepHypRef 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)
43olcs 875 . . 3 (((𝑅 ↾ {𝐴}) ∈ 𝑉 ∧ {𝐴} ∈ V) → (𝑅 “ {𝐴}) ∈ V)
52, 4mpan2 690 . 2 ((𝑅 ↾ {𝐴}) ∈ 𝑉 → (𝑅 “ {𝐴}) ∈ V)
61, 5eqeltrid 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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