![]() |
Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1256 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 34061. 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 |
---|---|
bnj1256.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1256.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1256.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1256.4 | ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ) |
bnj1256.5 | ⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} |
bnj1256.6 | ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))) |
bnj1256.7 | ⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥)) |
Ref | Expression |
---|---|
bnj1256 | ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1256.6 | . 2 ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))) | |
2 | abid 2713 | . . . 4 ⊢ (𝑔 ∈ {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉))} ↔ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉))) | |
3 | 2 | bnj1238 33805 | . . 3 ⊢ (𝑔 ∈ {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉))} → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑) |
4 | bnj1256.2 | . . . 4 ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
5 | bnj1256.3 | . . . 4 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
6 | eqid 2732 | . . . 4 ⊢ 〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉 = 〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
7 | eqid 2732 | . . . 4 ⊢ {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉))} = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉))} | |
8 | 4, 5, 6, 7 | bnj1234 34012 | . . 3 ⊢ 𝐶 = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉))} |
9 | 3, 8 | eleq2s 2851 | . 2 ⊢ (𝑔 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑) |
10 | 1, 9 | bnj770 33762 | 1 ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {cab 2709 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3432 ∩ cin 3946 ⊆ wss 3947 〈cop 4633 class class class wbr 5147 dom cdm 5675 ↾ cres 5677 Fn wfn 6535 ‘cfv 6540 ∧ w-bnj17 33685 predc-bnj14 33687 FrSe w-bnj15 33691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-res 5687 df-iota 6492 df-fun 6542 df-fn 6543 df-fv 6548 df-bnj17 33686 |
This theorem is referenced by: bnj1253 34016 bnj1286 34018 bnj1280 34019 |
Copyright terms: Public domain | W3C validator |