Theorem ereldm 6465

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 2207	. . . 4 ⊢ (𝜑 → (𝑥 ∈ [𝐴]𝑅 ↔ 𝑥 ∈ [𝐵]𝑅))
3	2	exbidv 1797	. . 3 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐵]𝑅))
4		ecdmn0m 6464	. . 3 ⊢ (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)
5		ecdmn0m 6464	. . 3 ⊢ (𝐵 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐵]𝑅)
6	3, 4, 5	3bitr4g 222	. 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ 𝐵 ∈ dom 𝑅))
7		ereldm.1	. . . 4 ⊢ (𝜑 → 𝑅 Er 𝑋)
8		erdm 6432	. . . 4 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
9	7, 8	syl 14	. . 3 ⊢ (𝜑 → dom 𝑅 = 𝑋)
10	9	eleq2d 2207	. 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ 𝐴 ∈ 𝑋))
11	9	eleq2d 2207	. 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐵 ∈ 𝑋))
12	6, 10, 11	3bitr3d 217	1 ⊢ (𝜑 → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  dom cdm 4534   Er wer 6419  [cec 6420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-er 6422  df-ec 6424

This theorem is referenced by:  erth  6466  brecop  6512