Theorem fodju0 7111

Description: Lemma for fodjuomni 7113 and fodjumkv 7124. A condition which shows that 𝐴 is empty. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)

Hypotheses

Ref	Expression
fodjuf.fo	⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))
fodjuf.p	⊢ 𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))
fodju0.1	⊢ (𝜑 → ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o)

Assertion

Ref	Expression
fodju0	⊢ (𝜑 → 𝐴 = ∅)

Distinct variable groups:   𝜑,𝑦,𝑧   𝑦,𝑂,𝑧   𝑧,𝐴   𝑧,𝐵   𝑧,𝐹   𝑦,𝐴   𝑦,𝐹   𝑤,𝑂   𝑤,𝑃

Allowed substitution hints:   𝜑(𝑤)   𝐴(𝑤)   𝐵(𝑦,𝑤)   𝑃(𝑦,𝑧)   𝐹(𝑤)

Proof of Theorem fodju0

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fodjuf.fo	. . . . 5 ⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))
2		djulcl 7016	. . . . 5 ⊢ (𝑢 ∈ 𝐴 → (inl‘𝑢) ∈ (𝐴 ⊔ 𝐵))
3		foelrn 5721	. . . . 5 ⊢ ((𝐹:𝑂–onto→(𝐴 ⊔ 𝐵) ∧ (inl‘𝑢) ∈ (𝐴 ⊔ 𝐵)) → ∃𝑣 ∈ 𝑂 (inl‘𝑢) = (𝐹‘𝑣))
4	1, 2, 3	syl2an 287	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → ∃𝑣 ∈ 𝑂 (inl‘𝑢) = (𝐹‘𝑣))
5		fodjuf.p	. . . . . 6 ⊢ 𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))
6		fveqeq2 5495	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑦) = (inl‘𝑧) ↔ (𝐹‘𝑣) = (inl‘𝑧)))
7	6	rexbidv 2467	. . . . . . 7 ⊢ (𝑦 = 𝑣 → (∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧)))
8	7	ifbid 3541	. . . . . 6 ⊢ (𝑦 = 𝑣 → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧), ∅, 1o))
9		simprl 521	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → 𝑣 ∈ 𝑂)
10		peano1 4571	. . . . . . . 8 ⊢ ∅ ∈ ω
11	10	a1i 9	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → ∅ ∈ ω)
12		1onn 6488	. . . . . . . 8 ⊢ 1o ∈ ω
13	12	a1i 9	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → 1o ∈ ω)
14	1	fodjuomnilemdc 7108	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑂) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧))
15	14	ad2ant2r 501	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧))
16	11, 13, 15	ifcldcd 3555	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧), ∅, 1o) ∈ ω)
17	5, 8, 9, 16	fvmptd3 5579	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → (𝑃‘𝑣) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧), ∅, 1o))
18		fveqeq2 5495	. . . . . 6 ⊢ (𝑤 = 𝑣 → ((𝑃‘𝑤) = 1o ↔ (𝑃‘𝑣) = 1o))
19		fodju0.1	. . . . . . 7 ⊢ (𝜑 → ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o)
20	19	ad2antrr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o)
21	18, 20, 9	rspcdva 2835	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → (𝑃‘𝑣) = 1o)
22		simplr 520	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → 𝑢 ∈ 𝐴)
23		simprr 522	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → (inl‘𝑢) = (𝐹‘𝑣))
24	23	eqcomd 2171	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → (𝐹‘𝑣) = (inl‘𝑢))
25		fveq2 5486	. . . . . . . 8 ⊢ (𝑧 = 𝑢 → (inl‘𝑧) = (inl‘𝑢))
26	25	rspceeqv 2848	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐴 ∧ (𝐹‘𝑣) = (inl‘𝑢)) → ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧))
27	22, 24, 26	syl2anc 409	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧))
28	27	iftrued 3527	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧), ∅, 1o) = ∅)
29	17, 21, 28	3eqtr3rd 2207	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → ∅ = 1o)
30	4, 29	rexlimddv 2588	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → ∅ = 1o)
31		1n0 6400	. . . . 5 ⊢ 1o ≠ ∅
32	31	nesymi 2382	. . . 4 ⊢ ¬ ∅ = 1o
33	32	a1i 9	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → ¬ ∅ = 1o)
34	30, 33	pm2.65da 651	. 2 ⊢ (𝜑 → ¬ 𝑢 ∈ 𝐴)
35	34	eq0rdv 3453	1 ⊢ (𝜑 → 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  DECID wdc 824   = wceq 1343   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445  ∅c0 3409  ifcif 3520   ↦ cmpt 4043  ωcom 4567  –onto→wfo 5186  ‘cfv 5188  1_oc1o 6377   ⊔ cdju 7002  inlcinl 7010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109  df-1o 6384  df-dju 7003  df-inl 7012  df-inr 7013

This theorem is referenced by:  fodjuomnilemres  7112  fodjumkvlemres  7123