Theorem ereldm 6574

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 2247	. . . 4 ⊢ (𝜑 → (𝑥 ∈ [𝐴]𝑅 ↔ 𝑥 ∈ [𝐵]𝑅))
3	2	exbidv 1825	. . 3 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐵]𝑅))
4		ecdmn0m 6573	. . 3 ⊢ (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)
5		ecdmn0m 6573	. . 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 6541	. . . 4 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
9	7, 8	syl 14	. . 3 ⊢ (𝜑 → dom 𝑅 = 𝑋)
10	9	eleq2d 2247	. 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ 𝐴 ∈ 𝑋))
11	9	eleq2d 2247	. 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 1353  ∃wex 1492   ∈ wcel 2148  dom cdm 4625   Er wer 6528  [cec 6529

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-xp 4631  df-cnv 4633  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-er 6531  df-ec 6533

This theorem is referenced by:  erth  6575  brecop  6621