Theorem ereldm 6664

Description: Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
ereldm.1	⊢ (𝜑 → 𝑅 Er 𝑋)
ereldm.2	⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

Assertion

Ref	Expression
ereldm	⊢ (𝜑 → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋))

Proof of Theorem ereldm

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ereldm.2	. . . . 5 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
2	1	eleq2d 2274	. . . 4 ⊢ (𝜑 → (𝑥 ∈ [𝐴]𝑅 ↔ 𝑥 ∈ [𝐵]𝑅))
3	2	exbidv 1847	. . 3 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐵]𝑅))
4		ecdmn0m 6663	. . 3 ⊢ (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)
5		ecdmn0m 6663	. . 3 ⊢ (𝐵 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐵]𝑅)
6	3, 4, 5	3bitr4g 223	. 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ 𝐵 ∈ dom 𝑅))
7		ereldm.1	. . . 4 ⊢ (𝜑 → 𝑅 Er 𝑋)
8		erdm 6629	. . . 4 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
9	7, 8	syl 14	. . 3 ⊢ (𝜑 → dom 𝑅 = 𝑋)
10	9	eleq2d 2274	. 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ 𝐴 ∈ 𝑋))
11	9	eleq2d 2274	. 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐵 ∈ 𝑋))
12	6, 10, 11	3bitr3d 218	1 ⊢ (𝜑 → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1372  ∃wex 1514   ∈ wcel 2175  dom cdm 4674   Er wer 6616  [cec 6617

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4680  df-cnv 4682  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-er 6619  df-ec 6621

This theorem is referenced by:  erth  6665  brecop  6711