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Theorem sprsymrelf1lem 46159
Description: Lemma for sprsymrelf1 46164. (Contributed by AV, 22-Nov-2021.)
Assertion
Ref Expression
sprsymrelf1lem ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎𝑏))
Distinct variable groups:   𝑉,𝑐   𝑎,𝑏,𝑐,𝑥,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem sprsymrelf1lem
Dummy variables 𝑝 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prssspr 46153 . . . . . 6 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑝𝑎) → ∃𝑖𝑉𝑗𝑉 𝑝 = {𝑖, 𝑗})
21ad4ant14 751 . . . . 5 ((((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}}) ∧ 𝑝𝑎) → ∃𝑖𝑉𝑗𝑉 𝑝 = {𝑖, 𝑗})
3 simpr 486 . . . . . . . . . . . 12 (((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑝 = {𝑖, 𝑗})
43adantr 482 . . . . . . . . . . 11 ((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}})) → 𝑝 = {𝑖, 𝑗})
54eleq1d 2819 . . . . . . . . . 10 ((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}})) → (𝑝𝑎 ↔ {𝑖, 𝑗} ∈ 𝑎))
6 simpr 486 . . . . . . . . . . . . . 14 ((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → {𝑖, 𝑗} ∈ 𝑎)
7 eqeq1 2737 . . . . . . . . . . . . . . 15 (𝑐 = {𝑖, 𝑗} → (𝑐 = {𝑖, 𝑗} ↔ {𝑖, 𝑗} = {𝑖, 𝑗}))
87adantl 483 . . . . . . . . . . . . . 14 (((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) ∧ 𝑐 = {𝑖, 𝑗}) → (𝑐 = {𝑖, 𝑗} ↔ {𝑖, 𝑗} = {𝑖, 𝑗}))
9 eqidd 2734 . . . . . . . . . . . . . 14 ((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → {𝑖, 𝑗} = {𝑖, 𝑗})
106, 8, 9rspcedvd 3615 . . . . . . . . . . . . 13 ((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → ∃𝑐𝑎 𝑐 = {𝑖, 𝑗})
1110adantlr 714 . . . . . . . . . . . 12 (((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}})) ∧ {𝑖, 𝑗} ∈ 𝑎) → ∃𝑐𝑎 𝑐 = {𝑖, 𝑗})
12 preq12 4740 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑖𝑦 = 𝑗) → {𝑥, 𝑦} = {𝑖, 𝑗})
1312eqeq2d 2744 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑖𝑦 = 𝑗) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑖, 𝑗}))
1413rexbidv 3179 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑖𝑦 = 𝑗) → (∃𝑐𝑎 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐𝑎 𝑐 = {𝑖, 𝑗}))
1514opelopabga 5534 . . . . . . . . . . . . . 14 ((𝑖𝑉𝑗𝑉) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐𝑎 𝑐 = {𝑖, 𝑗}))
1615bicomd 222 . . . . . . . . . . . . 13 ((𝑖𝑉𝑗𝑉) → (∃𝑐𝑎 𝑐 = {𝑖, 𝑗} ↔ ⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}}))
1716ad3antrrr 729 . . . . . . . . . . . 12 (((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}})) ∧ {𝑖, 𝑗} ∈ 𝑎) → (∃𝑐𝑎 𝑐 = {𝑖, 𝑗} ↔ ⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}}))
1811, 17mpbid 231 . . . . . . . . . . 11 (((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}})) ∧ {𝑖, 𝑗} ∈ 𝑎) → ⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}})
1918ex 414 . . . . . . . . . 10 ((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}})) → ({𝑖, 𝑗} ∈ 𝑎 → ⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}}))
205, 19sylbid 239 . . . . . . . . 9 ((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}})) → (𝑝𝑎 → ⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}}))
21 eleq2 2823 . . . . . . . . . . 11 ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} ↔ ⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}}))
2221ad2antll 728 . . . . . . . . . 10 ((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}})) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} ↔ ⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}}))
2313rexbidv 3179 . . . . . . . . . . . . 13 ((𝑥 = 𝑖𝑦 = 𝑗) → (∃𝑐𝑏 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐𝑏 𝑐 = {𝑖, 𝑗}))
2423opelopabga 5534 . . . . . . . . . . . 12 ((𝑖 ∈ V ∧ 𝑗 ∈ V) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐𝑏 𝑐 = {𝑖, 𝑗}))
2524el2v 3483 . . . . . . . . . . 11 (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐𝑏 𝑐 = {𝑖, 𝑗})
26 eqtr3 2759 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → 𝑝 = 𝑐)
2726equcomd 2023 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → 𝑐 = 𝑝)
2827eleq1d 2819 . . . . . . . . . . . . . . . . . . 19 ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → (𝑐𝑏𝑝𝑏))
2928biimpd 228 . . . . . . . . . . . . . . . . . 18 ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → (𝑐𝑏𝑝𝑏))
3029ex 414 . . . . . . . . . . . . . . . . 17 (𝑝 = {𝑖, 𝑗} → (𝑐 = {𝑖, 𝑗} → (𝑐𝑏𝑝𝑏)))
3130com13 88 . . . . . . . . . . . . . . . 16 (𝑐𝑏 → (𝑐 = {𝑖, 𝑗} → (𝑝 = {𝑖, 𝑗} → 𝑝𝑏)))
3231imp 408 . . . . . . . . . . . . . . 15 ((𝑐𝑏𝑐 = {𝑖, 𝑗}) → (𝑝 = {𝑖, 𝑗} → 𝑝𝑏))
3332rexlimiva 3148 . . . . . . . . . . . . . 14 (∃𝑐𝑏 𝑐 = {𝑖, 𝑗} → (𝑝 = {𝑖, 𝑗} → 𝑝𝑏))
3433com12 32 . . . . . . . . . . . . 13 (𝑝 = {𝑖, 𝑗} → (∃𝑐𝑏 𝑐 = {𝑖, 𝑗} → 𝑝𝑏))
3534adantl 483 . . . . . . . . . . . 12 (((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) → (∃𝑐𝑏 𝑐 = {𝑖, 𝑗} → 𝑝𝑏))
3635adantr 482 . . . . . . . . . . 11 ((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}})) → (∃𝑐𝑏 𝑐 = {𝑖, 𝑗} → 𝑝𝑏))
3725, 36biimtrid 241 . . . . . . . . . 10 ((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}})) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} → 𝑝𝑏))
3822, 37sylbid 239 . . . . . . . . 9 ((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}})) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} → 𝑝𝑏))
3920, 38syld 47 . . . . . . . 8 ((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}})) → (𝑝𝑎𝑝𝑏))
4039expimpd 455 . . . . . . 7 (((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) → ((((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}}) ∧ 𝑝𝑎) → 𝑝𝑏))
4140rexlimdva2 3158 . . . . . 6 (𝑖𝑉 → (∃𝑗𝑉 𝑝 = {𝑖, 𝑗} → ((((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}}) ∧ 𝑝𝑎) → 𝑝𝑏)))
4241rexlimiv 3149 . . . . 5 (∃𝑖𝑉𝑗𝑉 𝑝 = {𝑖, 𝑗} → ((((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}}) ∧ 𝑝𝑎) → 𝑝𝑏))
432, 42mpcom 38 . . . 4 ((((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}}) ∧ 𝑝𝑎) → 𝑝𝑏)
4443ex 414 . . 3 (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}}) → (𝑝𝑎𝑝𝑏))
4544ssrdv 3989 . 2 (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}}) → 𝑎𝑏)
4645ex 414 1 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎𝑏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wrex 3071  Vcvv 3475  wss 3949  {cpr 4631  cop 4635  {copab 5211  cfv 6544  Pairscspr 46145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-spr 46146
This theorem is referenced by:  sprsymrelf1  46164
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