Theorem ecexALTV 38289

Description: Existence of a coset, like ecexg 8769 but with a weaker antecedent: only the restriction of 𝑅 by the singleton of 𝐴 needs to be a set, not 𝑅 itself, see e.g. eccnvepex 38293. (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 8767	. 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
2		snex 5451	. . 3 ⊢ {𝐴} ∈ V
3		imaexALTV 38288	. . . 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 2848	1 ⊢ ((𝑅 ↾ {𝐴}) ∈ 𝑉 → [𝐴]𝑅 ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  Vcvv 3488  {csn 4648   ↾ cres 5702   “ cima 5703  [cec 8763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7772

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8767  df-qs 8771

This theorem is referenced by:  eccnvepex  38293