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Theorem erov 7789
Description: The value of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
eropr.1 𝐽 = (𝐴 / 𝑅)
eropr.2 𝐾 = (𝐵 / 𝑆)
eropr.3 (𝜑𝑇𝑍)
eropr.4 (𝜑𝑅 Er 𝑈)
eropr.5 (𝜑𝑆 Er 𝑉)
eropr.6 (𝜑𝑇 Er 𝑊)
eropr.7 (𝜑𝐴𝑈)
eropr.8 (𝜑𝐵𝑉)
eropr.9 (𝜑𝐶𝑊)
eropr.10 (𝜑+ :(𝐴 × 𝐵)⟶𝐶)
eropr.11 ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐵𝑢𝐵))) → ((𝑟𝑅𝑠𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))
eropr.12 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}
eropr.13 (𝜑𝑅𝑋)
eropr.14 (𝜑𝑆𝑌)
Assertion
Ref Expression
erov ((𝜑𝑃𝐴𝑄𝐵) → ([𝑃]𝑅 [𝑄]𝑆) = [(𝑃 + 𝑄)]𝑇)
Distinct variable groups:   𝑞,𝑝,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧,𝐴   𝐵,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝐽,𝑝,𝑞,𝑥,𝑦,𝑧   𝑃,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝑅,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝐾,𝑝,𝑞,𝑥,𝑦,𝑧   𝑄,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝑆,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   + ,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝜑,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝑇,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝑋,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑧   𝑌,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑧
Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   (𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝑈(𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝐽(𝑢,𝑡,𝑠,𝑟)   𝐾(𝑢,𝑡,𝑠,𝑟)   𝑉(𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝑊(𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)   𝑍(𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)

Proof of Theorem erov
StepHypRef Expression
1 eropr.1 . . . . 5 𝐽 = (𝐴 / 𝑅)
2 eropr.2 . . . . 5 𝐾 = (𝐵 / 𝑆)
3 eropr.3 . . . . 5 (𝜑𝑇𝑍)
4 eropr.4 . . . . 5 (𝜑𝑅 Er 𝑈)
5 eropr.5 . . . . 5 (𝜑𝑆 Er 𝑉)
6 eropr.6 . . . . 5 (𝜑𝑇 Er 𝑊)
7 eropr.7 . . . . 5 (𝜑𝐴𝑈)
8 eropr.8 . . . . 5 (𝜑𝐵𝑉)
9 eropr.9 . . . . 5 (𝜑𝐶𝑊)
10 eropr.10 . . . . 5 (𝜑+ :(𝐴 × 𝐵)⟶𝐶)
11 eropr.11 . . . . 5 ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐵𝑢𝐵))) → ((𝑟𝑅𝑠𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))
12 eropr.12 . . . . 5 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12erovlem 7788 . . . 4 (𝜑 = (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))
14133ad2ant1 1080 . . 3 ((𝜑𝑃𝐴𝑄𝐵) → = (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))
15 simprl 793 . . . . . . . 8 (((𝜑𝑃𝐴𝑄𝐵) ∧ (𝑥 = [𝑃]𝑅𝑦 = [𝑄]𝑆)) → 𝑥 = [𝑃]𝑅)
1615eqeq1d 2623 . . . . . . 7 (((𝜑𝑃𝐴𝑄𝐵) ∧ (𝑥 = [𝑃]𝑅𝑦 = [𝑄]𝑆)) → (𝑥 = [𝑝]𝑅 ↔ [𝑃]𝑅 = [𝑝]𝑅))
17 simprr 795 . . . . . . . 8 (((𝜑𝑃𝐴𝑄𝐵) ∧ (𝑥 = [𝑃]𝑅𝑦 = [𝑄]𝑆)) → 𝑦 = [𝑄]𝑆)
1817eqeq1d 2623 . . . . . . 7 (((𝜑𝑃𝐴𝑄𝐵) ∧ (𝑥 = [𝑃]𝑅𝑦 = [𝑄]𝑆)) → (𝑦 = [𝑞]𝑆 ↔ [𝑄]𝑆 = [𝑞]𝑆))
1916, 18anbi12d 746 . . . . . 6 (((𝜑𝑃𝐴𝑄𝐵) ∧ (𝑥 = [𝑃]𝑅𝑦 = [𝑄]𝑆)) → ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ↔ ([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆)))
2019anbi1d 740 . . . . 5 (((𝜑𝑃𝐴𝑄𝐵) ∧ (𝑥 = [𝑃]𝑅𝑦 = [𝑄]𝑆)) → (((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
21202rexbidv 3050 . . . 4 (((𝜑𝑃𝐴𝑄𝐵) ∧ (𝑥 = [𝑃]𝑅𝑦 = [𝑄]𝑆)) → (∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ∃𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
2221iotabidv 5831 . . 3 (((𝜑𝑃𝐴𝑄𝐵) ∧ (𝑥 = [𝑃]𝑅𝑦 = [𝑄]𝑆)) → (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) = (℩𝑧𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
23 eropr.13 . . . . 5 (𝜑𝑅𝑋)
24 ecelqsg 7747 . . . . . 6 ((𝑅𝑋𝑃𝐴) → [𝑃]𝑅 ∈ (𝐴 / 𝑅))
2524, 1syl6eleqr 2709 . . . . 5 ((𝑅𝑋𝑃𝐴) → [𝑃]𝑅𝐽)
2623, 25sylan 488 . . . 4 ((𝜑𝑃𝐴) → [𝑃]𝑅𝐽)
27263adant3 1079 . . 3 ((𝜑𝑃𝐴𝑄𝐵) → [𝑃]𝑅𝐽)
28 eropr.14 . . . . 5 (𝜑𝑆𝑌)
29 ecelqsg 7747 . . . . . 6 ((𝑆𝑌𝑄𝐵) → [𝑄]𝑆 ∈ (𝐵 / 𝑆))
3029, 2syl6eleqr 2709 . . . . 5 ((𝑆𝑌𝑄𝐵) → [𝑄]𝑆𝐾)
3128, 30sylan 488 . . . 4 ((𝜑𝑄𝐵) → [𝑄]𝑆𝐾)
32313adant2 1078 . . 3 ((𝜑𝑃𝐴𝑄𝐵) → [𝑄]𝑆𝐾)
33 iotaex 5827 . . . 4 (℩𝑧𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ V
3433a1i 11 . . 3 ((𝜑𝑃𝐴𝑄𝐵) → (℩𝑧𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ V)
3514, 22, 27, 32, 34ovmpt2d 6741 . 2 ((𝜑𝑃𝐴𝑄𝐵) → ([𝑃]𝑅 [𝑄]𝑆) = (℩𝑧𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
36 eqid 2621 . . . . . . 7 [𝑃]𝑅 = [𝑃]𝑅
37 eqid 2621 . . . . . . 7 [𝑄]𝑆 = [𝑄]𝑆
3836, 37pm3.2i 471 . . . . . 6 ([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆)
39 eqid 2621 . . . . . 6 [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇
4038, 39pm3.2i 471 . . . . 5 (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇)
41 eceq1 7727 . . . . . . . . 9 (𝑝 = 𝑃 → [𝑝]𝑅 = [𝑃]𝑅)
4241eqeq2d 2631 . . . . . . . 8 (𝑝 = 𝑃 → ([𝑃]𝑅 = [𝑝]𝑅 ↔ [𝑃]𝑅 = [𝑃]𝑅))
4342anbi1d 740 . . . . . . 7 (𝑝 = 𝑃 → (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ↔ ([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆)))
44 oveq1 6611 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝 + 𝑞) = (𝑃 + 𝑞))
4544eceq1d 7728 . . . . . . . 8 (𝑝 = 𝑃 → [(𝑝 + 𝑞)]𝑇 = [(𝑃 + 𝑞)]𝑇)
4645eqeq2d 2631 . . . . . . 7 (𝑝 = 𝑃 → ([(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇 ↔ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑞)]𝑇))
4743, 46anbi12d 746 . . . . . 6 (𝑝 = 𝑃 → ((([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇) ↔ (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑞)]𝑇)))
48 eceq1 7727 . . . . . . . . 9 (𝑞 = 𝑄 → [𝑞]𝑆 = [𝑄]𝑆)
4948eqeq2d 2631 . . . . . . . 8 (𝑞 = 𝑄 → ([𝑄]𝑆 = [𝑞]𝑆 ↔ [𝑄]𝑆 = [𝑄]𝑆))
5049anbi2d 739 . . . . . . 7 (𝑞 = 𝑄 → (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ↔ ([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆)))
51 oveq2 6612 . . . . . . . . 9 (𝑞 = 𝑄 → (𝑃 + 𝑞) = (𝑃 + 𝑄))
5251eceq1d 7728 . . . . . . . 8 (𝑞 = 𝑄 → [(𝑃 + 𝑞)]𝑇 = [(𝑃 + 𝑄)]𝑇)
5352eqeq2d 2631 . . . . . . 7 (𝑞 = 𝑄 → ([(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑞)]𝑇 ↔ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇))
5450, 53anbi12d 746 . . . . . 6 (𝑞 = 𝑄 → ((([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑞)]𝑇) ↔ (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇)))
5547, 54rspc2ev 3308 . . . . 5 ((𝑃𝐴𝑄𝐵 ∧ (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇)) → ∃𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇))
5640, 55mp3an3 1410 . . . 4 ((𝑃𝐴𝑄𝐵) → ∃𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇))
57563adant1 1077 . . 3 ((𝜑𝑃𝐴𝑄𝐵) → ∃𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇))
58 ecexg 7691 . . . . . 6 (𝑇𝑍 → [(𝑃 + 𝑄)]𝑇 ∈ V)
593, 58syl 17 . . . . 5 (𝜑 → [(𝑃 + 𝑄)]𝑇 ∈ V)
60593ad2ant1 1080 . . . 4 ((𝜑𝑃𝐴𝑄𝐵) → [(𝑃 + 𝑄)]𝑇 ∈ V)
61 simp1 1059 . . . . 5 ((𝜑𝑃𝐴𝑄𝐵) → 𝜑)
621, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11eroveu 7787 . . . . 5 ((𝜑 ∧ ([𝑃]𝑅𝐽 ∧ [𝑄]𝑆𝐾)) → ∃!𝑧𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
6361, 27, 32, 62syl12anc 1321 . . . 4 ((𝜑𝑃𝐴𝑄𝐵) → ∃!𝑧𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
64 simpr 477 . . . . . . 7 (((𝜑𝑃𝐴𝑄𝐵) ∧ 𝑧 = [(𝑃 + 𝑄)]𝑇) → 𝑧 = [(𝑃 + 𝑄)]𝑇)
6564eqeq1d 2623 . . . . . 6 (((𝜑𝑃𝐴𝑄𝐵) ∧ 𝑧 = [(𝑃 + 𝑄)]𝑇) → (𝑧 = [(𝑝 + 𝑞)]𝑇 ↔ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇))
6665anbi2d 739 . . . . 5 (((𝜑𝑃𝐴𝑄𝐵) ∧ 𝑧 = [(𝑃 + 𝑄)]𝑇) → ((([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇)))
67662rexbidv 3050 . . . 4 (((𝜑𝑃𝐴𝑄𝐵) ∧ 𝑧 = [(𝑃 + 𝑄)]𝑇) → (∃𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ∃𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇)))
6860, 63, 67iota2d 5835 . . 3 ((𝜑𝑃𝐴𝑄𝐵) → (∃𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇) ↔ (℩𝑧𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) = [(𝑃 + 𝑄)]𝑇))
6957, 68mpbid 222 . 2 ((𝜑𝑃𝐴𝑄𝐵) → (℩𝑧𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) = [(𝑃 + 𝑄)]𝑇)
7035, 69eqtrd 2655 1 ((𝜑𝑃𝐴𝑄𝐵) → ([𝑃]𝑅 [𝑄]𝑆) = [(𝑃 + 𝑄)]𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  ∃!weu 2469  wrex 2908  Vcvv 3186  wss 3555   class class class wbr 4613   × cxp 5072  cio 5808  wf 5843  (class class class)co 6604  {coprab 6605  cmpt2 6606   Er wer 7684  [cec 7685   / cqs 7686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-er 7687  df-ec 7689  df-qs 7693
This theorem is referenced by:  erov2  7791
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