Theorem ecexALTV 37835

Description: Existence of a coset, like ecexg 8735 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 37839. (Contributed by Peter Mazsa, 22-Feb-2023.)

Assertion

Ref	Expression
ecexALTV	⊢ ((𝑅 ↾ {𝐴}) ∈ 𝑉 → [𝐴]𝑅 ∈ V)

Proof of Theorem ecexALTV

Step	Hyp	Ref	Expression
1		df-ec 8733	. 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
2		snex 5437	. . 3 ⊢ {𝐴} ∈ V
3		imaexALTV 37834	. . . 4 ⊢ ((𝑅 ∈ V ∨ ((𝑅 ↾ {𝐴}) ∈ 𝑉 ∧ {𝐴} ∈ V)) → (𝑅 “ {𝐴}) ∈ V)
4	3	olcs 874	. . 3 ⊢ (((𝑅 ↾ {𝐴}) ∈ 𝑉 ∧ {𝐴} ∈ V) → (𝑅 “ {𝐴}) ∈ V)
5	2, 4	mpan2 689	. 2 ⊢ ((𝑅 ↾ {𝐴}) ∈ 𝑉 → (𝑅 “ {𝐴}) ∈ V)
6	1, 5	eqeltrid 2833	1 ⊢ ((𝑅 ↾ {𝐴}) ∈ 𝑉 → [𝐴]𝑅 ∈ V)

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2098  Vcvv 3473  {csn 4632   ↾ cres 5684   “ cima 5685  [cec 8729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ec 8733  df-qs 8737

This theorem is referenced by:  eccnvepex  37839