Theorem ecexALTV 36466

Description: Existence of a coset, like ecexg 8502 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 36470. (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 8500	. 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
2		snex 5354	. . 3 ⊢ {𝐴} ∈ V
3		imaexALTV 36465	. . . 4 ⊢ ((𝑅 ∈ V ∨ ((𝑅 ↾ {𝐴}) ∈ 𝑉 ∧ {𝐴} ∈ V)) → (𝑅 “ {𝐴}) ∈ V)
4	3	olcs 873	. . 3 ⊢ (((𝑅 ↾ {𝐴}) ∈ 𝑉 ∧ {𝐴} ∈ V) → (𝑅 “ {𝐴}) ∈ V)
5	2, 4	mpan2 688	. 2 ⊢ ((𝑅 ↾ {𝐴}) ∈ 𝑉 → (𝑅 “ {𝐴}) ∈ V)
6	1, 5	eqeltrid 2843	1 ⊢ ((𝑅 ↾ {𝐴}) ∈ 𝑉 → [𝐴]𝑅 ∈ V)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  Vcvv 3432  {csn 4561   ↾ cres 5591   “ cima 5592  [cec 8496

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ec 8500  df-qs 8504

This theorem is referenced by:  eccnvepex  36470