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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1256 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 32334. 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 2803 | . . . 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 32078 | . . 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 2821 | . . . 4 ⊢ 〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉 = 〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
7 | eqid 2821 | . . . 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 32285 | . . 3 ⊢ 𝐶 = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉))} |
9 | 3, 8 | eleq2s 2931 | . 2 ⊢ (𝑔 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑) |
10 | 1, 9 | bnj770 32034 | 1 ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 {cab 2799 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 ∩ cin 3935 ⊆ wss 3936 〈cop 4573 class class class wbr 5066 dom cdm 5555 ↾ cres 5557 Fn wfn 6350 ‘cfv 6355 ∧ w-bnj17 31956 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 |
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-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-br 5067 df-opab 5129 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 df-bnj17 31957 |
This theorem is referenced by: bnj1253 32289 bnj1286 32291 bnj1280 32292 |
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