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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj548 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj852 32193. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj548.1 | ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)) |
bnj548.2 | ⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) |
bnj548.3 | ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) |
bnj548.4 | ⊢ 𝐺 = (𝑓 ∪ {〈𝑚, 𝐶〉}) |
bnj548.5 | ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛) |
Ref | Expression |
---|---|
bnj548 | ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → 𝐵 = 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj548.5 | . . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛) | |
2 | 1 | bnj930 32041 | . . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → Fun 𝐺) |
3 | 2 | adantr 483 | . . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → Fun 𝐺) |
4 | bnj548.1 | . . . . . . . 8 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)) | |
5 | 4 | simp1bi 1141 | . . . . . . 7 ⊢ (𝜏 → 𝑓 Fn 𝑚) |
6 | fndm 6455 | . . . . . . . 8 ⊢ (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚) | |
7 | eleq2 2901 | . . . . . . . . 9 ⊢ (dom 𝑓 = 𝑚 → (𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ 𝑚)) | |
8 | 7 | biimpar 480 | . . . . . . . 8 ⊢ ((dom 𝑓 = 𝑚 ∧ 𝑖 ∈ 𝑚) → 𝑖 ∈ dom 𝑓) |
9 | 6, 8 | sylan 582 | . . . . . . 7 ⊢ ((𝑓 Fn 𝑚 ∧ 𝑖 ∈ 𝑚) → 𝑖 ∈ dom 𝑓) |
10 | 5, 9 | sylan 582 | . . . . . 6 ⊢ ((𝜏 ∧ 𝑖 ∈ 𝑚) → 𝑖 ∈ dom 𝑓) |
11 | 10 | 3ad2antl2 1182 | . . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → 𝑖 ∈ dom 𝑓) |
12 | 3, 11 | jca 514 | . . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → (Fun 𝐺 ∧ 𝑖 ∈ dom 𝑓)) |
13 | bnj548.4 | . . . . 5 ⊢ 𝐺 = (𝑓 ∪ {〈𝑚, 𝐶〉}) | |
14 | 13 | bnj931 32042 | . . . 4 ⊢ 𝑓 ⊆ 𝐺 |
15 | 12, 14 | jctil 522 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → (𝑓 ⊆ 𝐺 ∧ (Fun 𝐺 ∧ 𝑖 ∈ dom 𝑓))) |
16 | 3anan12 1092 | . . 3 ⊢ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝑖 ∈ dom 𝑓) ↔ (𝑓 ⊆ 𝐺 ∧ (Fun 𝐺 ∧ 𝑖 ∈ dom 𝑓))) | |
17 | 15, 16 | sylibr 236 | . 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → (Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝑖 ∈ dom 𝑓)) |
18 | funssfv 6691 | . 2 ⊢ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝑖 ∈ dom 𝑓) → (𝐺‘𝑖) = (𝑓‘𝑖)) | |
19 | iuneq1 4935 | . . . 4 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) | |
20 | 19 | eqcomd 2827 | . . 3 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
21 | bnj548.2 | . . 3 ⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) | |
22 | bnj548.3 | . . 3 ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) | |
23 | 20, 21, 22 | 3eqtr4g 2881 | . 2 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → 𝐵 = 𝐾) |
24 | 17, 18, 23 | 3syl 18 | 1 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → 𝐵 = 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∪ cun 3934 ⊆ wss 3936 {csn 4567 〈cop 4573 ∪ ciun 4919 dom cdm 5555 Fun wfun 6349 Fn wfn 6350 ‘cfv 6355 predc-bnj14 31958 FrSe w-bnj15 31962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-res 5567 df-iota 6314 df-fun 6357 df-fn 6358 df-fv 6363 |
This theorem is referenced by: bnj553 32170 |
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