Theorem ercpbllem 17462

Description: Lemma for ercpbl 17463. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by AV, 12-Jul-2024.)

Hypotheses

Ref	Expression
ercpbl.r	⊢ (𝜑 → ∼ Er 𝑉)
ercpbl.v	⊢ (𝜑 → 𝑉 ∈ 𝑊)
ercpbl.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
ercpbllem.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
ercpbllem	⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ 𝐴 ∼ 𝐵))

Distinct variable groups:   𝑥, ∼   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉   𝜑,𝑥

Allowed substitution hints:   𝐹(𝑥)   𝑊(𝑥)

Proof of Theorem ercpbllem

Step	Hyp	Ref	Expression
1		ercpbl.r	. . . 4 ⊢ (𝜑 → ∼ Er 𝑉)
2		ercpbl.v	. . . 4 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
3		ercpbl.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
4	1, 2, 3	divsfval 17461	. . 3 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )
5	1, 2, 3	divsfval 17461	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) = [𝐵] ∼ )
6	4, 5	eqeq12d 2749	. 2 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ [𝐴] ∼ = [𝐵] ∼ ))
7		ercpbllem.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8	1, 7	erth 8685	. 2 ⊢ (𝜑 → (𝐴 ∼ 𝐵 ↔ [𝐴] ∼ = [𝐵] ∼ ))
9	6, 8	bitr4d 282	1 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ 𝐴 ∼ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113   class class class wbr 5095   ↦ cmpt 5176  ‘cfv 6489   Er wer 8628  [cec 8629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fv 6497  df-er 8631  df-ec 8633

This theorem is referenced by:  ercpbl  17463  erlecpbl  17464