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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj548 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj852 30991. 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 30840 | . . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → Fun 𝐺) |
3 | 2 | adantr 481 | . . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → Fun 𝐺) |
4 | bnj548.1 | . . . . . . . 8 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)) | |
5 | 4 | simp1bi 1076 | . . . . . . 7 ⊢ (𝜏 → 𝑓 Fn 𝑚) |
6 | fndm 5990 | . . . . . . . 8 ⊢ (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚) | |
7 | eleq2 2690 | . . . . . . . . 9 ⊢ (dom 𝑓 = 𝑚 → (𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ 𝑚)) | |
8 | 7 | biimpar 502 | . . . . . . . 8 ⊢ ((dom 𝑓 = 𝑚 ∧ 𝑖 ∈ 𝑚) → 𝑖 ∈ dom 𝑓) |
9 | 6, 8 | sylan 488 | . . . . . . 7 ⊢ ((𝑓 Fn 𝑚 ∧ 𝑖 ∈ 𝑚) → 𝑖 ∈ dom 𝑓) |
10 | 5, 9 | sylan 488 | . . . . . 6 ⊢ ((𝜏 ∧ 𝑖 ∈ 𝑚) → 𝑖 ∈ dom 𝑓) |
11 | 10 | 3ad2antl2 1224 | . . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → 𝑖 ∈ dom 𝑓) |
12 | 3, 11 | jca 554 | . . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → (Fun 𝐺 ∧ 𝑖 ∈ dom 𝑓)) |
13 | bnj548.4 | . . . . 5 ⊢ 𝐺 = (𝑓 ∪ {〈𝑚, 𝐶〉}) | |
14 | 13 | bnj931 30841 | . . . 4 ⊢ 𝑓 ⊆ 𝐺 |
15 | 12, 14 | jctil 560 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → (𝑓 ⊆ 𝐺 ∧ (Fun 𝐺 ∧ 𝑖 ∈ dom 𝑓))) |
16 | 3anan12 1051 | . . 3 ⊢ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝑖 ∈ dom 𝑓) ↔ (𝑓 ⊆ 𝐺 ∧ (Fun 𝐺 ∧ 𝑖 ∈ dom 𝑓))) | |
17 | 15, 16 | sylibr 224 | . 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → (Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝑖 ∈ dom 𝑓)) |
18 | funssfv 6209 | . 2 ⊢ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝑖 ∈ dom 𝑓) → (𝐺‘𝑖) = (𝑓‘𝑖)) | |
19 | iuneq1 4534 | . . . 4 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) | |
20 | 19 | eqcomd 2628 | . . 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 2681 | . 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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∪ cun 3572 ⊆ wss 3574 {csn 4177 〈cop 4183 ∪ ciun 4520 dom cdm 5114 Fun wfun 5882 Fn wfn 5883 ‘cfv 5888 predc-bnj14 30754 FrSe w-bnj15 30758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-res 5126 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 |
This theorem is referenced by: bnj553 30968 |
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