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Theorem ecexALTV 35764
 Description: Existence of a coset, like ecexg 8278 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 35768. (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 8276 . 2 [𝐴]𝑅 = (𝑅 “ {𝐴})
2 snex 5297 . . 3 {𝐴} ∈ V
3 imaexALTV 35763 . . . 4 ((𝑅 ∈ V ∨ ((𝑅 ↾ {𝐴}) ∈ 𝑉 ∧ {𝐴} ∈ V)) → (𝑅 “ {𝐴}) ∈ V)
43olcs 873 . . 3 (((𝑅 ↾ {𝐴}) ∈ 𝑉 ∧ {𝐴} ∈ V) → (𝑅 “ {𝐴}) ∈ V)
52, 4mpan2 690 . 2 ((𝑅 ↾ {𝐴}) ∈ 𝑉 → (𝑅 “ {𝐴}) ∈ V)
61, 5eqeltrid 2894 1 ((𝑅 ↾ {𝐴}) ∈ 𝑉 → [𝐴]𝑅 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  Vcvv 3441  {csn 4525   ↾ cres 5521   “ cima 5522  [cec 8272 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ec 8276  df-qs 8280 This theorem is referenced by:  eccnvepex  35768
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