Theorem ereldm 6723

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 2299	. . . 4 ⊢ (𝜑 → (𝑥 ∈ [𝐴]𝑅 ↔ 𝑥 ∈ [𝐵]𝑅))
3	2	exbidv 1871	. . 3 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐵]𝑅))
4		ecdmn0m 6722	. . 3 ⊢ (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)
5		ecdmn0m 6722	. . 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 6688	. . . 4 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
9	7, 8	syl 14	. . 3 ⊢ (𝜑 → dom 𝑅 = 𝑋)
10	9	eleq2d 2299	. 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ 𝐴 ∈ 𝑋))
11	9	eleq2d 2299	. 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 1395  ∃wex 1538   ∈ wcel 2200  dom cdm 4718   Er wer 6675  [cec 6676

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4724  df-cnv 4726  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-er 6678  df-ec 6680

This theorem is referenced by:  erth  6724  brecop  6770