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Theorem recexprlemrnd 7749

Description: 𝐵 is rounded. Lemma for recexpr 7758. (Contributed by Jim Kingdon, 27-Dec-2019.)

**Hypothesis**
Ref	Expression
recexpr.1	⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩

**Assertion**
Ref	Expression
recexprlemrnd	⊢ (𝐴 ∈ P → (∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))))

Distinct variable groups: 𝑟,𝑞,𝑥,𝑦,𝐴 𝐵,𝑞,𝑟,𝑥,𝑦

**Proof of Theorem recexprlemrnd**
Step	Hyp	Ref	Expression
1		recexpr.1	. . . . . 6 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩
2	1	recexprlemopl 7745	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1^st ‘𝐵)) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)))
3	2	3expia 1208	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q) → (𝑞 ∈ (1^st ‘𝐵) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵))))
4	1	recexprlemlol 7746	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)) → 𝑞 ∈ (1^st ‘𝐵)))
5	3, 4	impbid 129	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q) → (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵))))
6	5	ralrimiva 2580	. 2 ⊢ (𝐴 ∈ P → ∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵))))
7	1	recexprlemopu 7747	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑟 ∈ Q ∧ 𝑟 ∈ (2^nd ‘𝐵)) → ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))
8	7	3expia 1208	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑟 ∈ Q) → (𝑟 ∈ (2^nd ‘𝐵) → ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵))))
9	1	recexprlemupu 7748	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)) → 𝑟 ∈ (2^nd ‘𝐵)))
10	8, 9	impbid 129	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝑟 ∈ Q) → (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵))))
11	10	ralrimiva 2580	. 2 ⊢ (𝐴 ∈ P → ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵))))
12	6, 11	jca 306	1 ⊢ (𝐴 ∈ P → (∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1373 ∃wex 1516 ∈ wcel 2177 {cab 2192 ∀wral 2485 ∃wrex 2486 ⟨cop 3637 class class class wbr 4047 ‘cfv 5276 1^st c1st 6231 2^nd c2nd 6232 Qcnq 7400 *_Qcrq 7404 <_Q cltq 7405 Pcnp 7411

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4163 ax-sep 4166 ax-nul 4174 ax-pow 4222 ax-pr 4257 ax-un 4484 ax-setind 4589 ax-iinf 4640

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-ral 2490 df-rex 2491 df-reu 2492 df-rab 2494 df-v 2775 df-sbc 3000 df-csb 3095 df-dif 3169 df-un 3171 df-in 3173 df-ss 3180 df-nul 3462 df-pw 3619 df-sn 3640 df-pr 3641 df-op 3643 df-uni 3853 df-int 3888 df-iun 3931 df-br 4048 df-opab 4110 df-mpt 4111 df-tr 4147 df-eprel 4340 df-id 4344 df-po 4347 df-iso 4348 df-iord 4417 df-on 4419 df-suc 4422 df-iom 4643 df-xp 4685 df-rel 4686 df-cnv 4687 df-co 4688 df-dm 4689 df-rn 4690 df-res 4691 df-ima 4692 df-iota 5237 df-fun 5278 df-fn 5279 df-f 5280 df-f1 5281 df-fo 5282 df-f1o 5283 df-fv 5284 df-ov 5954 df-oprab 5955 df-mpo 5956 df-1st 6233 df-2nd 6234 df-recs 6398 df-irdg 6463 df-1o 6509 df-oadd 6513 df-omul 6514 df-er 6627 df-ec 6629 df-qs 6633 df-ni 7424 df-pli 7425 df-mi 7426 df-lti 7427 df-plpq 7464 df-mpq 7465 df-enq 7467 df-nqqs 7468 df-plqqs 7469 df-mqqs 7470 df-1nqqs 7471 df-rq 7472 df-ltnqqs 7473

This theorem is referenced by: recexprlempr 7752

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