Theorem ercpbllemg 13627

Description: Lemma for ercpbl 13628. (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)
ercpbllem.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
ercpbllemg	⊢ (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 ercpbllemg

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		ercpbllem.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5	1, 2, 3, 4	divsfvalg 13626	. . 3 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )
6		ercpbllem.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7	1, 2, 3, 6	divsfvalg 13626	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) = [𝐵] ∼ )
8	5, 7	eqeq12d 2249	. 2 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ [𝐴] ∼ = [𝐵] ∼ ))
9	1, 4	erth 6826	. 2 ⊢ (𝜑 → (𝐴 ∼ 𝐵 ↔ [𝐴] ∼ = [𝐵] ∼ ))
10	8, 9	bitr4d 191	1 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ 𝐴 ∼ 𝐵))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398   ∈ wcel 2205   class class class wbr 4114   ↦ cmpt 4176  ‘cfv 5357   Er wer 6777  [cec 6778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fv 5365  df-er 6780  df-ec 6782

This theorem is referenced by:  ercpbl  13628  erlecpbl  13629