Theorem ercpbllem 17470

Description: Lemma for ercpbl 17471. (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 17469	. . 3 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )
5	1, 2, 3	divsfval 17469	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) = [𝐵] ∼ )
6	4, 5	eqeq12d 2745	. 2 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ [𝐴] ∼ = [𝐵] ∼ ))
7		ercpbllem.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8	1, 7	erth 8686	. 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 1540   ∈ wcel 2109   class class class wbr 5095   ↦ cmpt 5176  ‘cfv 6486   Er wer 8629  [cec 8630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  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 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fv 6494  df-er 8632  df-ec 8634

This theorem is referenced by:  ercpbl  17471  erlecpbl  17472