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Theorem ralxpmap 7793
 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 3176 . . 3 𝑔 ∈ V
2 snex 4835 . . 3 {⟨𝐽, 𝑦⟩} ∈ V
31, 2unex 6854 . 2 (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ V
4 simpr 476 . . . . . . 7 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝑓 ∈ (𝑆𝑚 𝑇))
5 elmapex 7764 . . . . . . . . 9 (𝑓 ∈ (𝑆𝑚 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
65adantl 481 . . . . . . . 8 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
7 elmapg 7757 . . . . . . . 8 ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝑓 ∈ (𝑆𝑚 𝑇) ↔ 𝑓:𝑇𝑆))
86, 7syl 17 . . . . . . 7 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → (𝑓 ∈ (𝑆𝑚 𝑇) ↔ 𝑓:𝑇𝑆))
94, 8mpbid 221 . . . . . 6 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝑓:𝑇𝑆)
10 simpl 472 . . . . . 6 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝐽𝑇)
119, 10ffvelrnd 6268 . . . . 5 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → (𝑓𝐽) ∈ 𝑆)
12 difss 3699 . . . . . . 7 (𝑇 ∖ {𝐽}) ⊆ 𝑇
13 fssres 5983 . . . . . . 7 ((𝑓:𝑇𝑆 ∧ (𝑇 ∖ {𝐽}) ⊆ 𝑇) → (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆)
149, 12, 13sylancl 693 . . . . . 6 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆)
155simpld 474 . . . . . . . 8 (𝑓 ∈ (𝑆𝑚 𝑇) → 𝑆 ∈ V)
1615adantl 481 . . . . . . 7 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝑆 ∈ V)
176simprd 478 . . . . . . . 8 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝑇 ∈ V)
18 difexg 4735 . . . . . . . 8 (𝑇 ∈ V → (𝑇 ∖ {𝐽}) ∈ V)
1917, 18syl 17 . . . . . . 7 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → (𝑇 ∖ {𝐽}) ∈ V)
2016, 19elmapd 7758 . . . . . 6 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})) ↔ (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆))
2114, 20mpbird 246 . . . . 5 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → (𝑓 ↾ (𝑇 ∖ {𝐽})) ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))
22 ffn 5958 . . . . . . 7 (𝑓:𝑇𝑆𝑓 Fn 𝑇)
239, 22syl 17 . . . . . 6 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝑓 Fn 𝑇)
24 fnsnsplit 6355 . . . . . 6 ((𝑓 Fn 𝑇𝐽𝑇) → 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓𝐽)⟩}))
2523, 10, 24syl2anc 691 . . . . 5 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓𝐽)⟩}))
26 opeq2 4341 . . . . . . . . 9 (𝑦 = (𝑓𝐽) → ⟨𝐽, 𝑦⟩ = ⟨𝐽, (𝑓𝐽)⟩)
2726sneqd 4137 . . . . . . . 8 (𝑦 = (𝑓𝐽) → {⟨𝐽, 𝑦⟩} = {⟨𝐽, (𝑓𝐽)⟩})
2827uneq2d 3729 . . . . . . 7 (𝑦 = (𝑓𝐽) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}) = (𝑔 ∪ {⟨𝐽, (𝑓𝐽)⟩}))
2928eqeq2d 2620 . . . . . 6 (𝑦 = (𝑓𝐽) → (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ↔ 𝑓 = (𝑔 ∪ {⟨𝐽, (𝑓𝐽)⟩})))
30 uneq1 3722 . . . . . . 7 (𝑔 = (𝑓 ↾ (𝑇 ∖ {𝐽})) → (𝑔 ∪ {⟨𝐽, (𝑓𝐽)⟩}) = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓𝐽)⟩}))
3130eqeq2d 2620 . . . . . 6 (𝑔 = (𝑓 ↾ (𝑇 ∖ {𝐽})) → (𝑓 = (𝑔 ∪ {⟨𝐽, (𝑓𝐽)⟩}) ↔ 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓𝐽)⟩})))
3229, 31rspc2ev 3295 . . . . 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 1318 . . . 4 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → ∃𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}))
3433ex 449 . . 3 (𝐽𝑇 → (𝑓 ∈ (𝑆𝑚 𝑇) → ∃𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩})))
35 elmapi 7765 . . . . . . . . . 10 (𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})) → 𝑔:(𝑇 ∖ {𝐽})⟶𝑆)
3635ad2antll 761 . . . . . . . . 9 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → 𝑔:(𝑇 ∖ {𝐽})⟶𝑆)
37 vex 3176 . . . . . . . . . . 11 𝑦 ∈ V
38 f1osng 6089 . . . . . . . . . . . 12 ((𝐽𝑇𝑦 ∈ V) → {⟨𝐽, 𝑦⟩}:{𝐽}–1-1-onto→{𝑦})
39 f1of 6050 . . . . . . . . . . . 12 ({⟨𝐽, 𝑦⟩}:{𝐽}–1-1-onto→{𝑦} → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
4038, 39syl 17 . . . . . . . . . . 11 ((𝐽𝑇𝑦 ∈ V) → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
4137, 40mpan2 703 . . . . . . . . . 10 (𝐽𝑇 → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
4241adantr 480 . . . . . . . . 9 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
43 incom 3767 . . . . . . . . . . 11 ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ({𝐽} ∩ (𝑇 ∖ {𝐽}))
44 disjdif 3992 . . . . . . . . . . 11 ({𝐽} ∩ (𝑇 ∖ {𝐽})) = ∅
4543, 44eqtri 2632 . . . . . . . . . 10 ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅
4645a1i 11 . . . . . . . . 9 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅)
47 fun 5979 . . . . . . . . 9 (((𝑔:(𝑇 ∖ {𝐽})⟶𝑆 ∧ {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦}) ∧ ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):((𝑇 ∖ {𝐽}) ∪ {𝐽})⟶(𝑆 ∪ {𝑦}))
4836, 42, 46, 47syl21anc 1317 . . . . . . . 8 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):((𝑇 ∖ {𝐽}) ∪ {𝐽})⟶(𝑆 ∪ {𝑦}))
49 uncom 3719 . . . . . . . . . 10 ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = ({𝐽} ∪ (𝑇 ∖ {𝐽}))
50 simpl 472 . . . . . . . . . . . 12 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → 𝐽𝑇)
5150snssd 4281 . . . . . . . . . . 11 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → {𝐽} ⊆ 𝑇)
52 undif 4001 . . . . . . . . . . 11 ({𝐽} ⊆ 𝑇 ↔ ({𝐽} ∪ (𝑇 ∖ {𝐽})) = 𝑇)
5351, 52sylib 207 . . . . . . . . . 10 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → ({𝐽} ∪ (𝑇 ∖ {𝐽})) = 𝑇)
5449, 53syl5eq 2656 . . . . . . . . 9 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = 𝑇)
5554feq2d 5944 . . . . . . . 8 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → ((𝑔 ∪ {⟨𝐽, 𝑦⟩}):((𝑇 ∖ {𝐽}) ∪ {𝐽})⟶(𝑆 ∪ {𝑦}) ↔ (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇⟶(𝑆 ∪ {𝑦})))
5648, 55mpbid 221 . . . . . . 7 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇⟶(𝑆 ∪ {𝑦}))
57 ssid 3587 . . . . . . . . 9 𝑆𝑆
5857a1i 11 . . . . . . . 8 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → 𝑆𝑆)
59 snssi 4280 . . . . . . . . 9 (𝑦𝑆 → {𝑦} ⊆ 𝑆)
6059ad2antrl 760 . . . . . . . 8 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → {𝑦} ⊆ 𝑆)
6158, 60unssd 3751 . . . . . . 7 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑆 ∪ {𝑦}) ⊆ 𝑆)
6256, 61fssd 5970 . . . . . 6 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇𝑆)
63 elmapex 7764 . . . . . . . . 9 (𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})) → (𝑆 ∈ V ∧ (𝑇 ∖ {𝐽}) ∈ V))
6463ad2antll 761 . . . . . . . 8 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑆 ∈ V ∧ (𝑇 ∖ {𝐽}) ∈ V))
6564simpld 474 . . . . . . 7 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → 𝑆 ∈ V)
66 ssun1 3738 . . . . . . . 8 𝑇 ⊆ (𝑇 ∪ {𝐽})
67 undif1 3995 . . . . . . . . 9 ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = (𝑇 ∪ {𝐽})
6864simprd 478 . . . . . . . . . 10 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑇 ∖ {𝐽}) ∈ V)
69 snex 4835 . . . . . . . . . 10 {𝐽} ∈ V
70 unexg 6857 . . . . . . . . . 10 (((𝑇 ∖ {𝐽}) ∈ V ∧ {𝐽} ∈ V) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) ∈ V)
7168, 69, 70sylancl 693 . . . . . . . . 9 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) ∈ V)
7267, 71syl5eqelr 2693 . . . . . . . 8 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑇 ∪ {𝐽}) ∈ V)
73 ssexg 4732 . . . . . . . 8 ((𝑇 ⊆ (𝑇 ∪ {𝐽}) ∧ (𝑇 ∪ {𝐽}) ∈ V) → 𝑇 ∈ V)
7466, 72, 73sylancr 694 . . . . . . 7 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → 𝑇 ∈ V)
7565, 74elmapd 7758 . . . . . 6 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → ((𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆𝑚 𝑇) ↔ (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇𝑆))
7662, 75mpbird 246 . . . . 5 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆𝑚 𝑇))
77 eleq1 2676 . . . . 5 (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → (𝑓 ∈ (𝑆𝑚 𝑇) ↔ (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆𝑚 𝑇)))
7876, 77syl5ibrcom 236 . . . 4 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → 𝑓 ∈ (𝑆𝑚 𝑇)))
7978rexlimdvva 3020 . . 3 (𝐽𝑇 → (∃𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → 𝑓 ∈ (𝑆𝑚 𝑇)))
8034, 79impbid 201 . 2 (𝐽𝑇 → (𝑓 ∈ (𝑆𝑚 𝑇) ↔ ∃𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩})))
81 ralxpmap.j . . 3 (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → (𝜑𝜓))
8281adantl 481 . 2 ((𝐽𝑇𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩})) → (𝜑𝜓))
833, 80, 82ralxpxfr2d 3298 1 (𝐽𝑇 → (∀𝑓 ∈ (𝑆𝑚 𝑇)𝜑 ↔ ∀𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽}))𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131   ↾ cres 5040   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746 This theorem is referenced by:  islindf4  19996
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