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Theorem ralxpmap 7949
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 3234 . . 3 𝑔 ∈ V
2 snex 4938 . . 3 {⟨𝐽, 𝑦⟩} ∈ V
31, 2unex 6998 . 2 (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ V
4 simpr 476 . . . . . . 7 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝑓 ∈ (𝑆𝑚 𝑇))
5 elmapex 7920 . . . . . . . . 9 (𝑓 ∈ (𝑆𝑚 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
65adantl 481 . . . . . . . 8 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
7 elmapg 7912 . . . . . . . 8 ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝑓 ∈ (𝑆𝑚 𝑇) ↔ 𝑓:𝑇𝑆))
86, 7syl 17 . . . . . . 7 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → (𝑓 ∈ (𝑆𝑚 𝑇) ↔ 𝑓:𝑇𝑆))
94, 8mpbid 222 . . . . . 6 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝑓:𝑇𝑆)
10 simpl 472 . . . . . 6 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝐽𝑇)
119, 10ffvelrnd 6400 . . . . 5 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → (𝑓𝐽) ∈ 𝑆)
12 difss 3770 . . . . . . 7 (𝑇 ∖ {𝐽}) ⊆ 𝑇
13 fssres 6108 . . . . . . 7 ((𝑓:𝑇𝑆 ∧ (𝑇 ∖ {𝐽}) ⊆ 𝑇) → (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆)
149, 12, 13sylancl 695 . . . . . 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 4841 . . . . . . . 8 (𝑇 ∈ V → (𝑇 ∖ {𝐽}) ∈ V)
1917, 18syl 17 . . . . . . 7 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → (𝑇 ∖ {𝐽}) ∈ V)
2016, 19elmapd 7913 . . . . . 6 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})) ↔ (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆))
2114, 20mpbird 247 . . . . 5 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → (𝑓 ↾ (𝑇 ∖ {𝐽})) ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))
22 ffn 6083 . . . . . . 7 (𝑓:𝑇𝑆𝑓 Fn 𝑇)
239, 22syl 17 . . . . . 6 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝑓 Fn 𝑇)
24 fnsnsplit 6491 . . . . . 6 ((𝑓 Fn 𝑇𝐽𝑇) → 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓𝐽)⟩}))
2523, 10, 24syl2anc 694 . . . . 5 ((𝐽𝑇𝑓 ∈ (𝑆𝑚 𝑇)) → 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓𝐽)⟩}))
26 opeq2 4434 . . . . . . . . 9 (𝑦 = (𝑓𝐽) → ⟨𝐽, 𝑦⟩ = ⟨𝐽, (𝑓𝐽)⟩)
2726sneqd 4222 . . . . . . . 8 (𝑦 = (𝑓𝐽) → {⟨𝐽, 𝑦⟩} = {⟨𝐽, (𝑓𝐽)⟩})
2827uneq2d 3800 . . . . . . 7 (𝑦 = (𝑓𝐽) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}) = (𝑔 ∪ {⟨𝐽, (𝑓𝐽)⟩}))
2928eqeq2d 2661 . . . . . 6 (𝑦 = (𝑓𝐽) → (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ↔ 𝑓 = (𝑔 ∪ {⟨𝐽, (𝑓𝐽)⟩})))
30 uneq1 3793 . . . . . . 7 (𝑔 = (𝑓 ↾ (𝑇 ∖ {𝐽})) → (𝑔 ∪ {⟨𝐽, (𝑓𝐽)⟩}) = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓𝐽)⟩}))
3130eqeq2d 2661 . . . . . 6 (𝑔 = (𝑓 ↾ (𝑇 ∖ {𝐽})) → (𝑓 = (𝑔 ∪ {⟨𝐽, (𝑓𝐽)⟩}) ↔ 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓𝐽)⟩})))
3229, 31rspc2ev 3355 . . . . 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 1366 . . . 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 7921 . . . . . . . . . 10 (𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})) → 𝑔:(𝑇 ∖ {𝐽})⟶𝑆)
3635ad2antll 765 . . . . . . . . 9 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → 𝑔:(𝑇 ∖ {𝐽})⟶𝑆)
37 vex 3234 . . . . . . . . . . 11 𝑦 ∈ V
38 f1osng 6215 . . . . . . . . . . . 12 ((𝐽𝑇𝑦 ∈ V) → {⟨𝐽, 𝑦⟩}:{𝐽}–1-1-onto→{𝑦})
39 f1of 6175 . . . . . . . . . . . 12 ({⟨𝐽, 𝑦⟩}:{𝐽}–1-1-onto→{𝑦} → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
4038, 39syl 17 . . . . . . . . . . 11 ((𝐽𝑇𝑦 ∈ V) → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
4137, 40mpan2 707 . . . . . . . . . 10 (𝐽𝑇 → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
4241adantr 480 . . . . . . . . 9 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
43 incom 3838 . . . . . . . . . . 11 ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ({𝐽} ∩ (𝑇 ∖ {𝐽}))
44 disjdif 4073 . . . . . . . . . . 11 ({𝐽} ∩ (𝑇 ∖ {𝐽})) = ∅
4543, 44eqtri 2673 . . . . . . . . . 10 ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅
4645a1i 11 . . . . . . . . 9 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅)
47 fun 6104 . . . . . . . . 9 (((𝑔:(𝑇 ∖ {𝐽})⟶𝑆 ∧ {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦}) ∧ ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):((𝑇 ∖ {𝐽}) ∪ {𝐽})⟶(𝑆 ∪ {𝑦}))
4836, 42, 46, 47syl21anc 1365 . . . . . . . 8 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):((𝑇 ∖ {𝐽}) ∪ {𝐽})⟶(𝑆 ∪ {𝑦}))
49 uncom 3790 . . . . . . . . . 10 ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = ({𝐽} ∪ (𝑇 ∖ {𝐽}))
50 simpl 472 . . . . . . . . . . . 12 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → 𝐽𝑇)
5150snssd 4372 . . . . . . . . . . 11 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → {𝐽} ⊆ 𝑇)
52 undif 4082 . . . . . . . . . . 11 ({𝐽} ⊆ 𝑇 ↔ ({𝐽} ∪ (𝑇 ∖ {𝐽})) = 𝑇)
5351, 52sylib 208 . . . . . . . . . 10 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → ({𝐽} ∪ (𝑇 ∖ {𝐽})) = 𝑇)
5449, 53syl5eq 2697 . . . . . . . . 9 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = 𝑇)
5554feq2d 6069 . . . . . . . 8 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → ((𝑔 ∪ {⟨𝐽, 𝑦⟩}):((𝑇 ∖ {𝐽}) ∪ {𝐽})⟶(𝑆 ∪ {𝑦}) ↔ (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇⟶(𝑆 ∪ {𝑦})))
5648, 55mpbid 222 . . . . . . 7 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇⟶(𝑆 ∪ {𝑦}))
57 ssid 3657 . . . . . . . . 9 𝑆𝑆
5857a1i 11 . . . . . . . 8 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → 𝑆𝑆)
59 snssi 4371 . . . . . . . . 9 (𝑦𝑆 → {𝑦} ⊆ 𝑆)
6059ad2antrl 764 . . . . . . . 8 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → {𝑦} ⊆ 𝑆)
6158, 60unssd 3822 . . . . . . 7 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑆 ∪ {𝑦}) ⊆ 𝑆)
6256, 61fssd 6095 . . . . . 6 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇𝑆)
63 elmapex 7920 . . . . . . . . 9 (𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})) → (𝑆 ∈ V ∧ (𝑇 ∖ {𝐽}) ∈ V))
6463ad2antll 765 . . . . . . . 8 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑆 ∈ V ∧ (𝑇 ∖ {𝐽}) ∈ V))
6564simpld 474 . . . . . . 7 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → 𝑆 ∈ V)
66 ssun1 3809 . . . . . . . 8 𝑇 ⊆ (𝑇 ∪ {𝐽})
67 undif1 4076 . . . . . . . . 9 ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = (𝑇 ∪ {𝐽})
6864simprd 478 . . . . . . . . . 10 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑇 ∖ {𝐽}) ∈ V)
69 snex 4938 . . . . . . . . . 10 {𝐽} ∈ V
70 unexg 7001 . . . . . . . . . 10 (((𝑇 ∖ {𝐽}) ∈ V ∧ {𝐽} ∈ V) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) ∈ V)
7168, 69, 70sylancl 695 . . . . . . . . 9 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) ∈ V)
7267, 71syl5eqelr 2735 . . . . . . . 8 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑇 ∪ {𝐽}) ∈ V)
73 ssexg 4837 . . . . . . . 8 ((𝑇 ⊆ (𝑇 ∪ {𝐽}) ∧ (𝑇 ∪ {𝐽}) ∈ V) → 𝑇 ∈ V)
7466, 72, 73sylancr 696 . . . . . . 7 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → 𝑇 ∈ V)
7565, 74elmapd 7913 . . . . . 6 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → ((𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆𝑚 𝑇) ↔ (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇𝑆))
7662, 75mpbird 247 . . . . 5 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆𝑚 𝑇))
77 eleq1 2718 . . . . 5 (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → (𝑓 ∈ (𝑆𝑚 𝑇) ↔ (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆𝑚 𝑇)))
7876, 77syl5ibrcom 237 . . . 4 ((𝐽𝑇 ∧ (𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽})))) → (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → 𝑓 ∈ (𝑆𝑚 𝑇)))
7978rexlimdvva 3067 . . 3 (𝐽𝑇 → (∃𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → 𝑓 ∈ (𝑆𝑚 𝑇)))
8034, 79impbid 202 . 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 3358 1 (𝐽𝑇 → (∀𝑓 ∈ (𝑆𝑚 𝑇)𝜑 ↔ ∀𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽}))𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210  cop 4216  cres 5145   Fn wfn 5921  wf 5922  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  𝑚 cmap 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901
This theorem is referenced by:  islindf4  20225
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