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Theorem ralxpmap 7770
Description: Quantification over functions in terms of quantification over values and punctured functions. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.)
Hypothesis
Ref Expression
ralxpmap.j (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → (𝜑𝜓))
Assertion
Ref Expression
ralxpmap (𝐽𝑇 → (∀𝑓 ∈ (𝑆𝑚 𝑇)𝜑 ↔ ∀𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽}))𝜓))
Distinct variable groups:   𝜑,𝑔,𝑦   𝜓,𝑓   𝑓,𝐽,𝑔,𝑦   𝑆,𝑓,𝑔,𝑦   𝑇,𝑓,𝑔,𝑦
Allowed substitution hints:   𝜑(𝑓)   𝜓(𝑦,𝑔)

Proof of Theorem ralxpmap
StepHypRef Expression
1 vex 3175 . . 3 𝑔 ∈ V
2 snex 4830 . . 3 {⟨𝐽, 𝑦⟩} ∈ V
31, 2unex 6831 . 2 (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ V
4 simpr 475 . . . . . . 7 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝑓 ∈ (𝑆𝑚 𝑇))
5 elmapex 7741 . . . . . . . . 9 (𝑓 ∈ (𝑆𝑚 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
65adantl 480 . . . . . . . 8 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
7 elmapg 7734 . . . . . . . 8 ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝑓 ∈ (𝑆𝑚 𝑇) ↔ 𝑓:𝑇𝑆))
86, 7syl 17 . . . . . . 7 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → (𝑓 ∈ (𝑆𝑚 𝑇) ↔ 𝑓:𝑇𝑆))
94, 8mpbid 220 . . . . . 6 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝑓:𝑇𝑆)
10 simpl 471 . . . . . 6 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝐽𝑇)
119, 10ffvelrnd 6253 . . . . 5 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → (𝑓𝐽) ∈ 𝑆)
12 difss 3698 . . . . . . 7 (𝑇 ∖ {𝐽}) ⊆ 𝑇
13 fssres 5968 . . . . . . 7 ((𝑓:𝑇𝑆 ∧ (𝑇 ∖ {𝐽}) ⊆ 𝑇) → (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆)
149, 12, 13sylancl 692 . . . . . 6 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆)
155simpld 473 . . . . . . . 8 (𝑓 ∈ (𝑆𝑚 𝑇) → 𝑆 ∈ V)
1615adantl 480 . . . . . . 7 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝑆 ∈ V)
176simprd 477 . . . . . . . 8 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝑇 ∈ V)
18 difexg 4730 . . . . . . . 8 (𝑇 ∈ V → (𝑇 ∖ {𝐽}) ∈ V)
1917, 18syl 17 . . . . . . 7 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → (𝑇 ∖ {𝐽}) ∈ V)
2016, 19elmapd 7735 . . . . . 6 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})) ↔ (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆))
2114, 20mpbird 245 . . . . 5 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → (𝑓 ↾ (𝑇 ∖ {𝐽})) ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))
22 ffn 5944 . . . . . . 7 (𝑓:𝑇𝑆𝑓 Fn 𝑇)
239, 22syl 17 . . . . . 6 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝑓 Fn 𝑇)
24 fnsnsplit 6333 . . . . . 6 ((𝑓 Fn 𝑇𝐽𝑇) → 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓𝐽)⟩}))
2523, 10, 24syl2anc 690 . . . . 5 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓𝐽)⟩}))
26 opeq2 4335 . . . . . . . . 9 (𝑦 = (𝑓𝐽) → ⟨𝐽, 𝑦⟩ = ⟨𝐽, (𝑓𝐽)⟩)
2726sneqd 4136 . . . . . . . 8 (𝑦 = (𝑓𝐽) → {⟨𝐽, 𝑦⟩} = {⟨𝐽, (𝑓𝐽)⟩})
2827uneq2d 3728 . . . . . . 7 (𝑦 = (𝑓𝐽) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}) = (𝑔 ∪ {⟨𝐽, (𝑓𝐽)⟩}))
2928eqeq2d 2619 . . . . . 6 (𝑦 = (𝑓𝐽) → (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ↔ 𝑓 = (𝑔 ∪ {⟨𝐽, (𝑓𝐽)⟩})))
30 uneq1 3721 . . . . . . 7 (𝑔 = (𝑓 ↾ (𝑇 ∖ {𝐽})) → (𝑔 ∪ {⟨𝐽, (𝑓𝐽)⟩}) = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓𝐽)⟩}))
3130eqeq2d 2619 . . . . . 6 (𝑔 = (𝑓 ↾ (𝑇 ∖ {𝐽})) → (𝑓 = (𝑔 ∪ {⟨𝐽, (𝑓𝐽)⟩}) ↔ 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓𝐽)⟩})))
3229, 31rspc2ev 3294 . . . . 5 (((𝑓𝐽) ∈ 𝑆 ∧ (𝑓 ↾ (𝑇 ∖ {𝐽})) ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})) ∧ 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓𝐽)⟩})) → ∃𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}))
3311, 21, 25, 32syl3anc 1317 . . . 4 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → ∃𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}))
3433ex 448 . . 3 (𝐽𝑇 → (𝑓 ∈ (𝑆𝑚 𝑇) → ∃𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩})))
35 elmapi 7742 . . . . . . . . . 10 (𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})) → 𝑔:(𝑇 ∖ {𝐽})⟶𝑆)
3635ad2antll 760 . . . . . . . . 9 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → 𝑔:(𝑇 ∖ {𝐽})⟶𝑆)
37 vex 3175 . . . . . . . . . . 11 𝑦 ∈ V
38 f1osng 6074 . . . . . . . . . . . 12 ((𝐽𝑇𝑦 ∈ V) → {⟨𝐽, 𝑦⟩}:{𝐽}–1-1-onto→{𝑦})
39 f1of 6035 . . . . . . . . . . . 12 ({⟨𝐽, 𝑦⟩}:{𝐽}–1-1-onto→{𝑦} → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
4038, 39syl 17 . . . . . . . . . . 11 ((𝐽𝑇𝑦 ∈ V) → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
4137, 40mpan2 702 . . . . . . . . . 10 (𝐽𝑇 → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
4241adantr 479 . . . . . . . . 9 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
43 incom 3766 . . . . . . . . . . 11 ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ({𝐽} ∩ (𝑇 ∖ {𝐽}))
44 disjdif 3991 . . . . . . . . . . 11 ({𝐽} ∩ (𝑇 ∖ {𝐽})) = ∅
4543, 44eqtri 2631 . . . . . . . . . 10 ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅
4645a1i 11 . . . . . . . . 9 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅)
47 fun 5965 . . . . . . . . 9 (((𝑔:(𝑇 ∖ {𝐽})⟶𝑆 ∧ {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦}) ∧ ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):((𝑇 ∖ {𝐽}) ∪ {𝐽})⟶(𝑆 ∪ {𝑦}))
4836, 42, 46, 47syl21anc 1316 . . . . . . . 8 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):((𝑇 ∖ {𝐽}) ∪ {𝐽})⟶(𝑆 ∪ {𝑦}))
49 uncom 3718 . . . . . . . . . 10 ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = ({𝐽} ∪ (𝑇 ∖ {𝐽}))
50 simpl 471 . . . . . . . . . . . 12 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → 𝐽𝑇)
5150snssd 4280 . . . . . . . . . . 11 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → {𝐽} ⊆ 𝑇)
52 undif 4000 . . . . . . . . . . 11 ({𝐽} ⊆ 𝑇 ↔ ({𝐽} ∪ (𝑇 ∖ {𝐽})) = 𝑇)
5351, 52sylib 206 . . . . . . . . . 10 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → ({𝐽} ∪ (𝑇 ∖ {𝐽})) = 𝑇)
5449, 53syl5eq 2655 . . . . . . . . 9 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = 𝑇)
5554feq2d 5930 . . . . . . . 8 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → ((𝑔 ∪ {⟨𝐽, 𝑦⟩}):((𝑇 ∖ {𝐽}) ∪ {𝐽})⟶(𝑆 ∪ {𝑦}) ↔ (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇⟶(𝑆 ∪ {𝑦})))
5648, 55mpbid 220 . . . . . . 7 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇⟶(𝑆 ∪ {𝑦}))
57 ssid 3586 . . . . . . . . 9 𝑆𝑆
5857a1i 11 . . . . . . . 8 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → 𝑆𝑆)
59 snssi 4279 . . . . . . . . 9 (𝑦𝑆 → {𝑦} ⊆ 𝑆)
6059ad2antrl 759 . . . . . . . 8 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → {𝑦} ⊆ 𝑆)
6158, 60unssd 3750 . . . . . . 7 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑆 ∪ {𝑦}) ⊆ 𝑆)
6256, 61fssd 5956 . . . . . 6 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇𝑆)
63 elmapex 7741 . . . . . . . . 9 (𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})) → (𝑆 ∈ V ∧ (𝑇 ∖ {𝐽}) ∈ V))
6463ad2antll 760 . . . . . . . 8 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑆 ∈ V ∧ (𝑇 ∖ {𝐽}) ∈ V))
6564simpld 473 . . . . . . 7 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → 𝑆 ∈ V)
66 ssun1 3737 . . . . . . . 8 𝑇 ⊆ (𝑇 ∪ {𝐽})
67 undif1 3994 . . . . . . . . 9 ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = (𝑇 ∪ {𝐽})
6864simprd 477 . . . . . . . . . 10 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑇 ∖ {𝐽}) ∈ V)
69 snex 4830 . . . . . . . . . 10 {𝐽} ∈ V
70 unexg 6834 . . . . . . . . . 10 (((𝑇 ∖ {𝐽}) ∈ V ∧ {𝐽} ∈ V) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) ∈ V)
7168, 69, 70sylancl 692 . . . . . . . . 9 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) ∈ V)
7267, 71syl5eqelr 2692 . . . . . . . 8 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑇 ∪ {𝐽}) ∈ V)
73 ssexg 4727 . . . . . . . 8 ((𝑇 ⊆ (𝑇 ∪ {𝐽}) ∧ (𝑇 ∪ {𝐽}) ∈ V) → 𝑇 ∈ V)
7466, 72, 73sylancr 693 . . . . . . 7 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → 𝑇 ∈ V)
7565, 74elmapd 7735 . . . . . 6 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → ((𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆𝑚 𝑇) ↔ (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇𝑆))
7662, 75mpbird 245 . . . . 5 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆𝑚 𝑇))
77 eleq1 2675 . . . . 5 (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → (𝑓 ∈ (𝑆𝑚 𝑇) ↔ (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆𝑚 𝑇)))
7876, 77syl5ibrcom 235 . . . 4 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → 𝑓 ∈ (𝑆𝑚 𝑇)))
7978rexlimdvva 3019 . . 3 (𝐽𝑇 → (∃𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → 𝑓 ∈ (𝑆𝑚 𝑇)))
8034, 79impbid 200 . 2 (𝐽𝑇 → (𝑓 ∈ (𝑆𝑚 𝑇) ↔ ∃𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩})))
81 ralxpmap.j . . 3 (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → (𝜑𝜓))
8281adantl 480 . 2 ((𝐽𝑇𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩})) → (𝜑𝜓))
833, 80, 82ralxpxfr2d 3297 1 (𝐽𝑇 → (∀𝑓 ∈ (𝑆𝑚 𝑇)𝜑 ↔ ∀𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽}))𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  wrex 2896  Vcvv 3172  cdif 3536  cun 3537  cin 3538  wss 3539  c0 3873  {csn 4124  cop 4130  cres 5030   Fn wfn 5785  wf 5786  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527  𝑚 cmap 7721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-map 7723
This theorem is referenced by:  islindf4  19938
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