Theorem ercpbllem 17554

Description: Lemma for ercpbl 17555. (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 17553	. . 3 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )
5	1, 2, 3	divsfval 17553	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) = [𝐵] ∼ )
6	4, 5	eqeq12d 2772	. 2 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ [𝐴] ∼ = [𝐵] ∼ ))
7		ercpbllem.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8	1, 7	erth 8721	. 2 ⊢ (𝜑 → (𝐴 ∼ 𝐵 ↔ [𝐴] ∼ = [𝐵] ∼ ))
9	6, 8	bitr4d 284	1 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ 𝐴 ∼ 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1554   ∈ wcel 2136   class class class wbr 5094   ↦ cmpt 5175  ‘cfv 6510   Er wer 8663  [cec 8664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6512  df-fv 6518  df-er 8666  df-ec 8668

This theorem is referenced by:  ercpbl  17555  erlecpbl  17556