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Theorem f1prex 7032
 Description: Relate a one-to-one function with a pair as domain and two different variables. (Contributed by Thierry Arnoux, 12-Jul-2020.)
Hypotheses
Ref Expression
f1prex.1 (𝑥 = (𝑓𝐴) → (𝜓𝜒))
f1prex.2 (𝑦 = (𝑓𝐵) → (𝜒𝜑))
Assertion
Ref Expression
f1prex ((𝐴𝑉𝐵𝑊𝐴𝐵) → (∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷𝜑) ↔ ∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝜓)))
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝐵,𝑓,𝑥,𝑦   𝐷,𝑓,𝑥,𝑦   𝑓,𝑉,𝑥,𝑦   𝑓,𝑊,𝑥,𝑦   𝜒,𝑥   𝜓,𝑓   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑓)   𝜓(𝑥,𝑦)   𝜒(𝑦,𝑓)

Proof of Theorem f1prex
StepHypRef Expression
1 simpl1 1188 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → 𝐴𝑉)
2 simpl2 1189 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → 𝐵𝑊)
3 simprl 770 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → 𝑓:{𝐴, 𝐵}–1-1𝐷)
4 f1f 6560 . . . . . . 7 (𝑓:{𝐴, 𝐵}–1-1𝐷𝑓:{𝐴, 𝐵}⟶𝐷)
53, 4syl 17 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → 𝑓:{𝐴, 𝐵}⟶𝐷)
6 fpr2g 6965 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → (𝑓:{𝐴, 𝐵}⟶𝐷 ↔ ((𝑓𝐴) ∈ 𝐷 ∧ (𝑓𝐵) ∈ 𝐷𝑓 = {⟨𝐴, (𝑓𝐴)⟩, ⟨𝐵, (𝑓𝐵)⟩})))
76biimpa 480 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ 𝑓:{𝐴, 𝐵}⟶𝐷) → ((𝑓𝐴) ∈ 𝐷 ∧ (𝑓𝐵) ∈ 𝐷𝑓 = {⟨𝐴, (𝑓𝐴)⟩, ⟨𝐵, (𝑓𝐵)⟩}))
87simp1d 1139 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ 𝑓:{𝐴, 𝐵}⟶𝐷) → (𝑓𝐴) ∈ 𝐷)
91, 2, 5, 8syl21anc 836 . . . . 5 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → (𝑓𝐴) ∈ 𝐷)
107simp2d 1140 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ 𝑓:{𝐴, 𝐵}⟶𝐷) → (𝑓𝐵) ∈ 𝐷)
111, 2, 5, 10syl21anc 836 . . . . 5 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → (𝑓𝐵) ∈ 𝐷)
12 prid1g 4653 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
131, 12syl 17 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → 𝐴 ∈ {𝐴, 𝐵})
14 prid2g 4654 . . . . . . . . 9 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
152, 14syl 17 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → 𝐵 ∈ {𝐴, 𝐵})
1613, 15jca 515 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → (𝐴 ∈ {𝐴, 𝐵} ∧ 𝐵 ∈ {𝐴, 𝐵}))
17 simpl3 1190 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → 𝐴𝐵)
18 f1veqaeq 7007 . . . . . . . . 9 ((𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝐴 ∈ {𝐴, 𝐵} ∧ 𝐵 ∈ {𝐴, 𝐵})) → ((𝑓𝐴) = (𝑓𝐵) → 𝐴 = 𝐵))
1918necon3d 2972 . . . . . . . 8 ((𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝐴 ∈ {𝐴, 𝐵} ∧ 𝐵 ∈ {𝐴, 𝐵})) → (𝐴𝐵 → (𝑓𝐴) ≠ (𝑓𝐵)))
2019imp 410 . . . . . . 7 (((𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝐴 ∈ {𝐴, 𝐵} ∧ 𝐵 ∈ {𝐴, 𝐵})) ∧ 𝐴𝐵) → (𝑓𝐴) ≠ (𝑓𝐵))
213, 16, 17, 20syl21anc 836 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → (𝑓𝐴) ≠ (𝑓𝐵))
22 simprr 772 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → 𝜑)
2321, 22jca 515 . . . . 5 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → ((𝑓𝐴) ≠ (𝑓𝐵) ∧ 𝜑))
24 neeq1 3013 . . . . . . 7 (𝑥 = (𝑓𝐴) → (𝑥𝑦 ↔ (𝑓𝐴) ≠ 𝑦))
25 f1prex.1 . . . . . . 7 (𝑥 = (𝑓𝐴) → (𝜓𝜒))
2624, 25anbi12d 633 . . . . . 6 (𝑥 = (𝑓𝐴) → ((𝑥𝑦𝜓) ↔ ((𝑓𝐴) ≠ 𝑦𝜒)))
27 neeq2 3014 . . . . . . 7 (𝑦 = (𝑓𝐵) → ((𝑓𝐴) ≠ 𝑦 ↔ (𝑓𝐴) ≠ (𝑓𝐵)))
28 f1prex.2 . . . . . . 7 (𝑦 = (𝑓𝐵) → (𝜒𝜑))
2927, 28anbi12d 633 . . . . . 6 (𝑦 = (𝑓𝐵) → (((𝑓𝐴) ≠ 𝑦𝜒) ↔ ((𝑓𝐴) ≠ (𝑓𝐵) ∧ 𝜑)))
3026, 29rspc2ev 3553 . . . . 5 (((𝑓𝐴) ∈ 𝐷 ∧ (𝑓𝐵) ∈ 𝐷 ∧ ((𝑓𝐴) ≠ (𝑓𝐵) ∧ 𝜑)) → ∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝜓))
319, 11, 23, 30syl3anc 1368 . . . 4 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)) → ∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝜓))
3231ex 416 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ((𝑓:{𝐴, 𝐵}–1-1𝐷𝜑) → ∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝜓)))
3332exlimdv 1934 . 2 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷𝜑) → ∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝜓)))
34 simpll1 1209 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → 𝐴𝑉)
35 simplrl 776 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → 𝑥𝐷)
3634, 35jca 515 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → (𝐴𝑉𝑥𝐷))
37 simpll2 1210 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → 𝐵𝑊)
38 simplrr 777 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → 𝑦𝐷)
3937, 38jca 515 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → (𝐵𝑊𝑦𝐷))
40 simpll3 1211 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → 𝐴𝐵)
41 simprl 770 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → 𝑥𝑦)
42 f1oprg 6646 . . . . . . . . . 10 (((𝐴𝑉𝑥𝐷) ∧ (𝐵𝑊𝑦𝐷)) → ((𝐴𝐵𝑥𝑦) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1-onto→{𝑥, 𝑦}))
4342imp 410 . . . . . . . . 9 ((((𝐴𝑉𝑥𝐷) ∧ (𝐵𝑊𝑦𝐷)) ∧ (𝐴𝐵𝑥𝑦)) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1-onto→{𝑥, 𝑦})
4436, 39, 40, 41, 43syl22anc 837 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1-onto→{𝑥, 𝑦})
45 f1of1 6601 . . . . . . . 8 ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1-onto→{𝑥, 𝑦} → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1→{𝑥, 𝑦})
4644, 45syl 17 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1→{𝑥, 𝑦})
4735, 38prssd 4712 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → {𝑥, 𝑦} ⊆ 𝐷)
48 f1ss 6566 . . . . . . 7 (({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1→{𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝐷) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1𝐷)
4946, 47, 48syl2anc 587 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1𝐷)
50 fvpr1g 6945 . . . . . . . 8 ((𝐴𝑉𝑥𝐷𝐴𝐵) → ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐴) = 𝑥)
5150eqcomd 2764 . . . . . . 7 ((𝐴𝑉𝑥𝐷𝐴𝐵) → 𝑥 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐴))
5234, 35, 40, 51syl3anc 1368 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → 𝑥 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐴))
53 fvpr2g 6946 . . . . . . . 8 ((𝐵𝑊𝑦𝐷𝐴𝐵) → ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
5453eqcomd 2764 . . . . . . 7 ((𝐵𝑊𝑦𝐷𝐴𝐵) → 𝑦 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐵))
5537, 38, 40, 54syl3anc 1368 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → 𝑦 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐵))
56 prex 5301 . . . . . . 7 {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} ∈ V
57 f1eq1 6555 . . . . . . . 8 (𝑓 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → (𝑓:{𝐴, 𝐵}–1-1𝐷 ↔ {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1𝐷))
58 fveq1 6657 . . . . . . . . . 10 (𝑓 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → (𝑓𝐴) = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐴))
5958eqeq2d 2769 . . . . . . . . 9 (𝑓 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → (𝑥 = (𝑓𝐴) ↔ 𝑥 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐴)))
60 fveq1 6657 . . . . . . . . . 10 (𝑓 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → (𝑓𝐵) = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐵))
6160eqeq2d 2769 . . . . . . . . 9 (𝑓 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → (𝑦 = (𝑓𝐵) ↔ 𝑦 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐵)))
6259, 61anbi12d 633 . . . . . . . 8 (𝑓 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → ((𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)) ↔ (𝑥 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐴) ∧ 𝑦 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐵))))
6357, 62anbi12d 633 . . . . . . 7 (𝑓 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → ((𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵))) ↔ ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐴) ∧ 𝑦 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐵)))))
6456, 63spcev 3525 . . . . . 6 (({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐴) ∧ 𝑦 = ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}‘𝐵))) → ∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵))))
6549, 52, 55, 64syl12anc 835 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → ∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵))))
66 simprl 770 . . . . . . . 8 (((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)))) → 𝑓:{𝐴, 𝐵}–1-1𝐷)
67 simplrr 777 . . . . . . . . . 10 (((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)))) → 𝜓)
68 simprrl 780 . . . . . . . . . . 11 (((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)))) → 𝑥 = (𝑓𝐴))
6968, 25syl 17 . . . . . . . . . 10 (((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)))) → (𝜓𝜒))
7067, 69mpbid 235 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)))) → 𝜒)
71 simprrr 781 . . . . . . . . . 10 (((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)))) → 𝑦 = (𝑓𝐵))
7271, 28syl 17 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)))) → (𝜒𝜑))
7370, 72mpbid 235 . . . . . . . 8 (((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)))) → 𝜑)
7466, 73jca 515 . . . . . . 7 (((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) ∧ (𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵)))) → (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑))
7574ex 416 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → ((𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵))) → (𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)))
7675eximdv 1918 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → (∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷 ∧ (𝑥 = (𝑓𝐴) ∧ 𝑦 = (𝑓𝐵))) → ∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)))
7765, 76mpd 15 . . . 4 ((((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑥𝑦𝜓)) → ∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷𝜑))
7877ex 416 . . 3 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝑥𝐷𝑦𝐷)) → ((𝑥𝑦𝜓) → ∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)))
7978rexlimdvva 3218 . 2 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝜓) → ∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷𝜑)))
8033, 79impbid 215 1 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷𝜑) ↔ ∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2951  ∃wrex 3071   ⊆ wss 3858  {cpr 4524  ⟨cop 4528  ⟶wf 6331  –1-1→wf1 6332  –1-1-onto→wf1o 6334  ‘cfv 6335 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343 This theorem is referenced by:  istrkg3ld  26354
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