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Theorem sprsymrelf1lem 43482
Description: Lemma for sprsymrelf1 43487. (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 43476 . . . . . 6 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑝𝑎) → ∃𝑖𝑉𝑗𝑉 𝑝 = {𝑖, 𝑗})
21ad4ant14 748 . . . . 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 485 . . . . . . . . . . . 12 (((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑝 = {𝑖, 𝑗})
43adantr 481 . . . . . . . . . . 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 2901 . . . . . . . . . 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 485 . . . . . . . . . . . . . 14 ((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → {𝑖, 𝑗} ∈ 𝑎)
7 eqeq1 2828 . . . . . . . . . . . . . . 15 (𝑐 = {𝑖, 𝑗} → (𝑐 = {𝑖, 𝑗} ↔ {𝑖, 𝑗} = {𝑖, 𝑗}))
87adantl 482 . . . . . . . . . . . . . 14 (((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) ∧ 𝑐 = {𝑖, 𝑗}) → (𝑐 = {𝑖, 𝑗} ↔ {𝑖, 𝑗} = {𝑖, 𝑗}))
9 eqidd 2825 . . . . . . . . . . . . . 14 ((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → {𝑖, 𝑗} = {𝑖, 𝑗})
106, 8, 9rspcedvd 3629 . . . . . . . . . . . . 13 ((((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → ∃𝑐𝑎 𝑐 = {𝑖, 𝑗})
1110adantlr 711 . . . . . . . . . . . 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 4669 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑖𝑦 = 𝑗) → {𝑥, 𝑦} = {𝑖, 𝑗})
1312eqeq2d 2835 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑖𝑦 = 𝑗) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑖, 𝑗}))
1413rexbidv 3301 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑖𝑦 = 𝑗) → (∃𝑐𝑎 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐𝑎 𝑐 = {𝑖, 𝑗}))
1514opelopabga 5416 . . . . . . . . . . . . . 14 ((𝑖𝑉𝑗𝑉) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐𝑎 𝑐 = {𝑖, 𝑗}))
1615bicomd 224 . . . . . . . . . . . . 13 ((𝑖𝑉𝑗𝑉) → (∃𝑐𝑎 𝑐 = {𝑖, 𝑗} ↔ ⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}}))
1716ad3antrrr 726 . . . . . . . . . . . 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 233 . . . . . . . . . . 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 413 . . . . . . . . . 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 241 . . . . . . . . 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 2905 . . . . . . . . . . 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 725 . . . . . . . . . 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 3301 . . . . . . . . . . . . 13 ((𝑥 = 𝑖𝑦 = 𝑗) → (∃𝑐𝑏 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐𝑏 𝑐 = {𝑖, 𝑗}))
2423opelopabga 5416 . . . . . . . . . . . 12 ((𝑖 ∈ V ∧ 𝑗 ∈ V) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐𝑏 𝑐 = {𝑖, 𝑗}))
2524el2v 3506 . . . . . . . . . . 11 (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐𝑏 𝑐 = {𝑖, 𝑗})
26 eqtr3 2847 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → 𝑝 = 𝑐)
2726equcomd 2019 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → 𝑐 = 𝑝)
2827eleq1d 2901 . . . . . . . . . . . . . . . . . . 19 ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → (𝑐𝑏𝑝𝑏))
2928biimpd 230 . . . . . . . . . . . . . . . . . 18 ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → (𝑐𝑏𝑝𝑏))
3029ex 413 . . . . . . . . . . . . . . . . 17 (𝑝 = {𝑖, 𝑗} → (𝑐 = {𝑖, 𝑗} → (𝑐𝑏𝑝𝑏)))
3130com13 88 . . . . . . . . . . . . . . . 16 (𝑐𝑏 → (𝑐 = {𝑖, 𝑗} → (𝑝 = {𝑖, 𝑗} → 𝑝𝑏)))
3231imp 407 . . . . . . . . . . . . . . 15 ((𝑐𝑏𝑐 = {𝑖, 𝑗}) → (𝑝 = {𝑖, 𝑗} → 𝑝𝑏))
3332rexlimiva 3285 . . . . . . . . . . . . . 14 (∃𝑐𝑏 𝑐 = {𝑖, 𝑗} → (𝑝 = {𝑖, 𝑗} → 𝑝𝑏))
3433com12 32 . . . . . . . . . . . . 13 (𝑝 = {𝑖, 𝑗} → (∃𝑐𝑏 𝑐 = {𝑖, 𝑗} → 𝑝𝑏))
3534adantl 482 . . . . . . . . . . . 12 (((𝑖𝑉𝑗𝑉) ∧ 𝑝 = {𝑖, 𝑗}) → (∃𝑐𝑏 𝑐 = {𝑖, 𝑗} → 𝑝𝑏))
3635adantr 481 . . . . . . . . . . 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, 36syl5bi 243 . . . . . . . . . 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 241 . . . . . . . . 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 454 . . . . . . 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 3291 . . . . . 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 3284 . . . . 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 413 . . 3 (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}}) → (𝑝𝑎𝑝𝑏))
4544ssrdv 3976 . 2 (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}}) → 𝑎𝑏)
4645ex 413 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 207  wa 396   = wceq 1530  wcel 2106  wrex 3143  Vcvv 3499  wss 3939  {cpr 4565  cop 4569  {copab 5124  cfv 6351  Pairscspr 43468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-spr 43469
This theorem is referenced by:  sprsymrelf1  43487
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