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Theorem sprsymrelf1lem 47416
Description: Lemma for sprsymrelf1 47421. (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 47410 . . . . . 6 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑝𝑎) → ∃𝑖𝑉𝑗𝑉 𝑝 = {𝑖, 𝑗})
21ad4ant14 752 . . . . 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 484 . . . . . . . . . . . 12 (((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑝 = {𝑖, 𝑗})
43adantr 480 . . . . . . . . . . 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 2824 . . . . . . . . . 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 484 . . . . . . . . . . . . . 14 ((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → {𝑖, 𝑗} ∈ 𝑎)
7 eqeq1 2739 . . . . . . . . . . . . . . 15 (𝑐 = {𝑖, 𝑗} → (𝑐 = {𝑖, 𝑗} ↔ {𝑖, 𝑗} = {𝑖, 𝑗}))
87adantl 481 . . . . . . . . . . . . . 14 (((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) ∧ 𝑐 = {𝑖, 𝑗}) → (𝑐 = {𝑖, 𝑗} ↔ {𝑖, 𝑗} = {𝑖, 𝑗}))
9 eqidd 2736 . . . . . . . . . . . . . 14 ((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → {𝑖, 𝑗} = {𝑖, 𝑗})
106, 8, 9rspcedvd 3624 . . . . . . . . . . . . 13 ((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → ∃𝑐𝑎 𝑐 = {𝑖, 𝑗})
1110adantlr 715 . . . . . . . . . . . 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 2746 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑖𝑦 = 𝑗) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑖, 𝑗}))
1413rexbidv 3177 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑖𝑦 = 𝑗) → (∃𝑐𝑎 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐𝑎 𝑐 = {𝑖, 𝑗}))
1514opelopabga 5543 . . . . . . . . . . . . . 14 ((𝑖𝑉𝑗𝑉) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐𝑎 𝑐 = {𝑖, 𝑗}))
1615bicomd 223 . . . . . . . . . . . . 13 ((𝑖𝑉𝑗𝑉) → (∃𝑐𝑎 𝑐 = {𝑖, 𝑗} ↔ ⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}}))
1716ad3antrrr 730 . . . . . . . . . . . 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 232 . . . . . . . . . . 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 412 . . . . . . . . . 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 240 . . . . . . . . 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 2828 . . . . . . . . . . 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 729 . . . . . . . . . 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 3177 . . . . . . . . . . . . 13 ((𝑥 = 𝑖𝑦 = 𝑗) → (∃𝑐𝑏 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐𝑏 𝑐 = {𝑖, 𝑗}))
2423opelopabga 5543 . . . . . . . . . . . 12 ((𝑖 ∈ V ∧ 𝑗 ∈ V) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐𝑏 𝑐 = {𝑖, 𝑗}))
2524el2v 3485 . . . . . . . . . . 11 (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐𝑏 𝑐 = {𝑖, 𝑗})
26 eqtr3 2761 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → 𝑝 = 𝑐)
2726equcomd 2016 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → 𝑐 = 𝑝)
2827eleq1d 2824 . . . . . . . . . . . . . . . . . . 19 ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → (𝑐𝑏𝑝𝑏))
2928biimpd 229 . . . . . . . . . . . . . . . . . 18 ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → (𝑐𝑏𝑝𝑏))
3029ex 412 . . . . . . . . . . . . . . . . 17 (𝑝 = {𝑖, 𝑗} → (𝑐 = {𝑖, 𝑗} → (𝑐𝑏𝑝𝑏)))
3130com13 88 . . . . . . . . . . . . . . . 16 (𝑐𝑏 → (𝑐 = {𝑖, 𝑗} → (𝑝 = {𝑖, 𝑗} → 𝑝𝑏)))
3231imp 406 . . . . . . . . . . . . . . 15 ((𝑐𝑏𝑐 = {𝑖, 𝑗}) → (𝑝 = {𝑖, 𝑗} → 𝑝𝑏))
3332rexlimiva 3145 . . . . . . . . . . . . . 14 (∃𝑐𝑏 𝑐 = {𝑖, 𝑗} → (𝑝 = {𝑖, 𝑗} → 𝑝𝑏))
3433com12 32 . . . . . . . . . . . . 13 (𝑝 = {𝑖, 𝑗} → (∃𝑐𝑏 𝑐 = {𝑖, 𝑗} → 𝑝𝑏))
3534adantl 481 . . . . . . . . . . . 12 (((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) → (∃𝑐𝑏 𝑐 = {𝑖, 𝑗} → 𝑝𝑏))
3635adantr 480 . . . . . . . . . . 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 242 . . . . . . . . . 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 240 . . . . . . . . 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 453 . . . . . . 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 3155 . . . . . 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 3146 . . . . 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 412 . . 3 (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}}) → (𝑝𝑎𝑝𝑏))
4544ssrdv 4001 . 2 (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}}) → 𝑎𝑏)
4645ex 412 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 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478  wss 3963  {cpr 4633  cop 4637  {copab 5210  cfv 6563  Pairscspr 47402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-spr 47403
This theorem is referenced by:  sprsymrelf1  47421
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