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 30384. 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 2598 | . . . 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 30131 | . . 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 2610 | . . . 4 ⊢ 〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉 = 〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
7 | eqid 2610 | . . . 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 30335 | . . 3 ⊢ 𝐶 = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉))} |
9 | 3, 8 | eleq2s 2706 | . 2 ⊢ (𝑔 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑) |
10 | 1, 9 | bnj770 30087 | 1 ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {cab 2596 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 ∩ cin 3539 ⊆ wss 3540 〈cop 4131 class class class wbr 4583 dom cdm 5038 ↾ cres 5040 Fn wfn 5799 ‘cfv 5804 ∧ w-bnj17 30005 predc-bnj14 30007 FrSe w-bnj15 30011 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
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-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-bnj17 30006 |
This theorem is referenced by: bnj1253 30339 bnj1286 30341 bnj1280 30342 |
Copyright terms: Public domain | W3C validator |